eSign Word Computer

What does productivity mean to you? We’re all expert procrastinators whether we like to admit it or not, the digital age has created an abundance of distractions that can keep you from focusing on important things, all day long if it comes down to it. Even the subject of productivity tools comes down to a single factor — it’s all designed for mental discipline training, because even though these tools exist and they’re useful to be more disciplined at your work, they’ll only be as good as the user himself.For: All-In-One OrganizationProofHubProofHub lets you stay in ultimate control of your projects, remote teams and clients even when you are on the go. Aside from giving team member the ability to manage and collaborate projects in real-time, the other big reason for choosing ProofHub is its super simple and brings everything under one roof. ProofHub is as simple as you want it to be (clean UI, to-do lists, project planning, group chat, review deign files, project reporting), but also as advanced as they sometimes need it to be (assign custom roles, white labeling. multilingual, time tracking, private tasks, complex multi-milestone projects, large remote team, n number of tasks).For: Social Media MastermindsBufferBuffer makes it extraordinarily easy to share web articles via social media without headache and hassle. Simply log into Buffer, connect social media accounts, and schedule specific content sharing times. When you’re out and about surfing the web and see an article you know your followers will love, hit the Buffer bookmarklet button and Buffer will throw that article into your queue, which it will then share at one of your designated share times.Buffer helps promote great content across the web and helps you store and share content so you don’t cram articles down the throats of your followers all at once.IFTTTIFTTT, standing for If This Then That, lets you craft “recipes” that allow for various apps and services to connect and work together. For example, I can create an IFTTT recipe that automatically uploads my Instagram photos to my Google Drive account. Create awesome integration with zero programming experience! If you can think it, you can IFTTT it.IFTTT can be utilized in endless combinations, and not just with regards to social media. However, social media users will find this an essential tool in their social shed.For: CopywritingODesk / ELanceSometimes the best man for the job is someone who is not you. Heck, probably pretty often the best man for the job isn’t you. Maybe because you aren’t even a man; maybe you’re a carrot. Who knows?If you need to get going with blogging content (and trust me, you do, because everyone does), then ODesk and ELance are decent places to start.These sites are free to sign up with, then you provide them with 10% of each payment per project. Alternatively, scour LinkedIn for freelancers. Usually you can get a decent assessment of a freelancer’s skill level by checking out their profile and doing a little digging. Whichever path you take, just remember that copy can’t be done single-handedly, so don’t be afraid to hire out some help.HemingwayBack when I was compiling a list of the best content marketing tools, one WordStream commenter (thanks Nicole!) made a suggestion of trying out Hemingway – it’s a great little tool that checks your grammar and highlights potential sentence danger zones as you write. Give it a type!For: News & Blog JunkiesPodkickerIf you’re a podcast addict, you’ll want a podcast app to help you organize your subscriptions. I’ve been using Podkicker for years and really enjoy its simple, easy-to-use interface. It’s free, but you can also update to Podkicker Pro (to get rid of ads) for $2.PocketPocket is a sweet little app that lets you save awesome content (articles, images, videos, etc.) to check out later.Picture this: you’re surfing around researching and see a fun article about the all-time best Pokemon (Arcanine, duh). Hit the Pocket button in your bookmark toolbar and the article will go straight to your Pocket. Pocket syncs across devices so you that later on you can read those fun articles and watch YouTube videos of mini pot-bellied pigs during the train ride home.

My TOP list is as follows:1. Communication tool - SlackOur team has been successfully using Skype for quite a while, and this is a way out for some teams, both small and bigger ones, but Slack is more convenient to use for IT teams, for marketers, sales people, and other industry-specific teams.2. Time tracking tool - ClockifyI like Clockify – it’s very simple, you create a task, you start tracking your time, you stop when you’re done, AND you can also check how much time you spent weekly on each working task.3. To do list – Trello boardsThe next tool I LOVE is Trello. This is a perfect one for outlining your day to day tasks, your future tasks, mapping out your ideas, sharing them with your team members, and so on.4. G Suite.No comments - couldn’t do without it.5. CalendlyIF you have a large number of meetings daily.6. Project management toolThroughout my years in IT I have met and used different PM tools, like Jira, YouTrack, Asana, and TeamGantt. And I must say that each one is good for its own purpose – so you simply choose the one that work best for you.7. Reporting tool - ExcelI have not used any specific reporting tool: I use Excel tables, both on Google drive and offline Excel files.8. CanvaNo comments :) Saves a fortune on a designer for those SMW owners whose budget is limited.See the full version of my tips here:8 BEST Tools for Remote Professionals

LIST OF 5 BEST BUSINESS BOOKS1)RICH DAD-POOR DAD,BY ROBERT KIYOSAKIRich Dad Poor Dad is a 1997 book written by Robert Kiyosaki and Sharon Lechter. It advocates the importance of financial literacy, financial independence and building wealth through investing in assets.TOP 3 RICH DAD POOR DAD QUOTES1-“In school we learn that mistakes are bad, and we are punished for making them. Yet, if you look at the way humans are designed to learn, we learn by making mistakes. We learn to walk by falling down. If we never fell down, we would never walk.”2-“Winners are not afraid of losing. But losers are. Failure is part of the process of success. People who avoid failure also avoid success.”3-“You’re only poor if you give up. The most important thing is that you did something. Most people only talk and dream of getting rich. You’ve done something.”2)THE SNOWBALL,BY ALICE SCHROEDERThe Snowball: Warren Buffett and the Business of Life is a biography of Warren Buffett by Alice SchroederTOP 3 SNOWBALL:WARREN BUFFETT AND BUSINESS OF LIFE QUOTES1-“On me personally what has been the most important was to understand the value of time -- and this is something that has come from observing him, learning his story and that time compounds. What you do when you are young (and as you use time over your life) can have an exponential effect so that if you are thoughtful about it, you can really have powerful results later, if you want to.2-“Time is the friend of the wonderful business, the enemy of the mediocre.”3-“Everybody wants attention and admiration. Nobody wants to be criticized. The sweetest sound in the English language is the sound of a person’s own name. The only way to get the best of an argument is to avoid it. If you are wrong, admit it quickly and emphatically. Ask questions instead of giving direct orders. Give the other person a fine reputation to live up to. Call attention to people’s mistakes indirectly. Let the other person save face.”3)ZERO TO ONE,BY PETER THIELZero to One: Notes on Startups, or How to Build the Future is a 2014 book by venture capitalist, PayPal co-founder, and early Facebook investor Peter Thiel along with Blake MastersIf you want to build a better future, you must believe in secrets.The great secret of our time is that there are still uncharted frontiers to explore and new inventions to create. In Zero to One , legendary entrepreneur and investor Peter Thiel shows how we can find singular ways to create those new things.TOP 3 ZERO TO ONE QUOTES1-ZERO TO ONE EVERY MOMENT IN BUSINESS happens only once. The next Bill Gates will not build an operating system. The next Larry Page or Sergey Brin won’t make a search engine. And the next Mark Zuckerberg won’t create a social network. If you are copying these guys, you aren’t learning from them.2-“The best entrepreneurs know this: every great business is built around a secret that’s hidden from the outside. A great company is a conspiracy to change the world; when you share your secret, the recipient becomes a fellow conspirator.”3-“Elite students climb confidently until they signNow a level of competition sufficiently intense to beat their dreams out of them. Higher education is the place where people who had big plans in high school get stuck in fierce rivalries with equally smart peers over conventional careers like management consulting and investment banking. For the privilege of being turned into conformists, students (or their families) pay hundreds of thousands of dollars in skyrocketing tuition that continues to outpace inflation. Why are we doing this to ourselves?4)STEVE JOBS,BY WALTER ISAACSONThe book is no doubt the best. It's inspiringIt will urge to do something in your life rather fulfilling it for some other person's dream.TOP 3 STEVE JOBS BIOGRAPHY QUOTES1-“One way to remember who you are is to remember who your heroes are.”2-“If you want to live your life in a creative way, as an artist, you have to not look back too much. You have to be willing to take whatever you’ve done and whoever you were and throw them away. The more the outside world tries to reinforce an image of you, the harder it is to continue to be an artist, which is why a lot of times, artists have to say, “Bye. I have to go. I’m going crazy and I’m getting out of here.” And they go and hibernate somewhere. Maybe later they re-emerge a little differently. (Steve Jobs)”3-“If you act like you can do something, then it will work)5)PRINCIPLES,BY RAY DALIORay Dalio, one of the world's most successful investors and entrepreneurs, shares the unconventional principles that he's developed, refined, and used over the past forty years to create unique results in both life and business—and which any person or organization can adopt to help achieve their goalsTOP 3 PRINCIPLES QUOTES1-“Truth - more precisely, an accurate understanding of reality - is the essential foundation for producing good outcomes.”2-“Nature gave us pain as a messaging device to tell us that we are approaching, or that we have exceeded, our limits in some way.”3-“It is far more common for people to allow ego to stand in the way of learning.”

Because Identity Politics got real tedious.Leftist Identity Politics is made up of the following rules:Everyone is oppressed by a mysterious thing called The White Patriarchy.Since actual oppression is hard to find, oppression now includes tiny little version of oppression called “microaggressions.”People can be oppressed by microaggressions because they’re incredibly fragile.Since The White Patriarchy oppresses everyone, all the oppressed people deserve extra rights and privileges to “level the playing field.”The Left has gone bonkers with Identity Politics. Currently, “I am oppressed by the White Patriarchy” is the default argument for EVERYTHING.We are bombarded with constant complaints of micro-oppression and constant virtue signalling. This stuff is non-stop now. Everyone is defiantly protesting their fake oppression.Here are French Resistance fighters in WWII fighting the German occupation:By contrast, here’s someone who is taking on the White Patriarchy for insisting that she has hair. I think the idea is that without the White Patriarchy forcing women to have hair, women would shave their head.I don’t know. But look at the expression of defiance on her face. Take THAT, caucasian males!Here are two people who are anti-fascist. Just like those French Resistance fighters from WWII! The difference is that instead of living in Nazi occupied France, they live in a rock-solid Constitutional Republic, with a Bill of Rights that guarantees their personal freedom.And their president, for good or bad, is a political outsider who was hated and attacked by his own party. Yet the voters made him president anyway. That’s a textbook example of democracy in action.It’s almost like they’re people who have declared everyone they disagree with to be a “fascist” because it’s dramatic sounding, and they lack the mental capacity to come up with a valid political argument. And it’s self-aggrandizing.Here’s a empowered woman who is really jazzed up about how a female imaginary person saved the life of a male imaginary person.Oh no you did dint! (z snap!) Take that, White Patriarchy!Here’s a woman who is protesting against being forced to go back to the 1950s.Looks like she forgot to put the word “the” before “1950s.” So she had to write it in later with someone’s Bic felt-tip pen. Might want to lay off the Zinfandel until after the sign is done next time.Take THAT, White Patriarchy!So now we learn that we’re supposed to fight the Gestapo by using this word “Latinix.”Why People Are Using The Term 'Latinx'“Latinx is the gender-neutral alternative to Latino, Latina and even Latin@. Used by scholars, activists and an increasing number of journalists, Latinx is quickly gaining popularity among the general public. It’s part of a “linguistic revolution” that aims to move beyond gender binaries and is inclusive of the intersecting identities of Latin American descendants.”Take THAT, White Patriarchy!I learned German in high school. German has gendered words too. For example, “Der Löffel” (spoon) is masculine, but “Die Gabel” (fork) is feminine, and “Das Messer” (knife) is neuter.This must be stopped! Help your silverware fight the Gestapo!What about YOU?Are you brave enough to fight the Gestapo by using the word “latinix” and by carrying signs about Harry Potter for some unspecified reason? COME ON, BE A HERO.Comments are welcome. I will not block you, so write whatever you like.But please don’t use “gendered words.” And don’t grill two pieces of corn and two hot dogs side by side with a bunch of toxic men. Also, wait until the charcoal gets hot enough before you start cooking. That way, it won’t smoke so much.Also, I would appreciate it if no micro, pico or nano aggressions showed up in my comments section. Use this chart for reference:

Because we spell it as it should be. Ligatures, like ‘æ’ and ‘œ’ are pronounced differently to ‘a’, ‘e’, or ‘o’ on their own… and slightly differently from ‘ae’ and ‘oe’ as separate letters. In fact, ‘æ’ and ‘œ’are considered as separate individual letters, in the same way as accented and modified letters (like à, á, â, é, è, ï, ø, ô, ß, ü, ñ, ç, č, š, etc.) are in other languages… some of which have made their way into English as in ‘naïve’, café, ‘resumé’ (to differentiate it from ‘resume’) and ‘clientèle’.US English seems to be particularly lazy about spelling different words differently (we’ll use a ‘cheque’ to pay for something, but we ‘check’ the quality of the item - I hope we’re all educated enough describe someone born in Prague or Brno as being Czech.)I’m not sure whether America modified the spelling to match a common mispronunciation of words, or whether the American pronunciation followed the changes in spelling made by that illiterate idiot, Webster. - For example ‘pædophile’ is pronounced in America as ‘ped’ (to rhyme with red) ‘o’ ‘file’. Elsewhere, the ‘æ’ is pronounced as a sound halfway between a short ‘a’ and a short ‘e’ (separately written, as ‘ae’, the paired letters should be pronounced as halfway between a long ‘a’ and a ‘long ‘e’). The ‘œ’ ligature and ‘oe’ pairing follows the same guidelines. ‘Fœtus’ or ‘fetus’, and ‘mediæval’ or ‘medieval’ are other examples.How do Americans spell the few words adopted into English which are given different spellings to reflect their gender? (as many words are in other languages)… I refer to words like ‘fiancé’ (male) and ‘fiancée’ (female), or ‘blond’ (male), and ‘blonde’ (female). Does American English differentiate?

Sure it can.Both people and computers can represent numbers in a whole lot of ways. The word “binary” usually refers to how you represent small natural numbers in base [math]2[/math], but there are many different representation schemes for other types of numbers.What you may be thinking of is that the rational number [math]1/10[/math] cannot be precisely represented by one of those representation schemes: floating-point arithmetic with a base-[math]2[/math] exponent.In floating-point arithmetic with base [math]b[/math] we represent a number [math]X[/math] as a triplet of numbers [math](s,m,e)[/math] where[math]\displaystyle X=(-1)^s\times m \times b^e[/math]Here [math]s[/math] is just [math]0[/math] (for positive numbers) or [math]1[/math] (for negative numbers), [math]m[/math] is called a mantissa and is a natural number in some fixed range, [math]b[/math] is a fixed number (usually [math]2[/math] or [math]10[/math]) and [math]e[/math] is another (positive or negative) integer called the exponent, also limited to some prescribed range.This is a pretty good representation scheme. It combines high precision with a decent range for both large and small numbers, and it enables arithmetic operations to be carried out reasonably efficiently – though it makes addition and subtraction much harder than multiplication and division, confusing many people who are used to it being the other way around.As you can see, there are different flavors of this representation, depending on the value of [math]b[/math] and the allowed ranges for [math]m[/math] and [math]e[/math]. One of the prevailing current standards, IEEE 754, doesn't actually enforce all of those details. It allows both [math]b=2[/math] and [math]b=10[/math].If [math]b=10[/math], you can definitely represent [math]1/10[/math] precisely:[math]\displaystyle \frac{1}{10}=(-1)^0\times 1\times 10^{-1}[/math]so [math]1/10[/math] is represented with complete precision as [math](0,1,-1)[/math].This, incidentally, is totally independent of how the numbers [math]m[/math] and [math]e[/math] themselves are represented. They can be stored as base [math]2[/math], base [math]10[/math] or anything else; the sign of [math]e[/math] can be stored separately, or sometime a fixed offset is subtracted from the stored value of [math]e[/math], and so on – it doesn't matter. As long as you’re using [math]b=10[/math], you can represent [math]1/10[/math] precisely.Now, if you choose to implement floating-point arithmetic using [math]b=2[/math] as a base, then of course the only numbers you can represent precisely are certain rational numbers whose denominators are a power of [math]2[/math]. You certainly can't precisely represent anything else, including fractions whose denominator has prime factors other than [math]2[/math].But again: floating-point arithmetic with base [math]2[/math] isn't the only option for floating-point arithmetic, and floating-point arithmetic of any kind isn't the only way of representing numbers in computers. Computer hardware or software can be designed to represent fractions [math]a/b[/math] as pairs of integers [math](a,b)[/math], allowing you to represent any rational number whose numerator and denominator are within a reasonable range. In this method, [math]1/10[/math] would get stored as [math](1,10)[/math], which is [math](1,1010)[/math] in binary. Not a problem at all.

Contrary to a lot of answers here, I don't immediately condemn it.Some people use unusual, advanced, or complex vocabulary because that's what they are familiar with. Often, they don't even realise that their own vocabulary might be harder to understand for some people. I refer to this as ‘the slang of the posh’, they’re words that aren't in everyday usage by most people simply because we have different (more colloquial, plain, or ‘working class’) terms to describe the same phenomena. Examples could be a more posh person using the term ‘drawing room’ where I would plump for the term ‘living room’. This is perfectly understandable, different people are familiar with different terms, and so long as I am not judged for the words I choose to use I wouldn't dream of judging another for the terms they feel more comfortable with.Some people may also use ‘big words’ because it's technical jargon. Usually, we create specific words for things we don't want to keep describing. In physics, I would use ‘quasar’ or ‘supermassive’ to mean a thing that would otherwise require description, despite the fact other people might be unfamiliar with what I'm referring to. It applies everywhere. In terms of archery, I could refer to a ‘riser’, or ‘clicker’ and someone who isn't familiar with archery would have to ask what I'm talking about. Jargon can get extremely complicated, usually moreso the more a person is involved in a particular thing. Now an important thing to note about jargon is often after a while you forget it is such; you subconsciously expect others to know what you mean, because to you it actually is an everyday term. If you encounter someone using jargon you don't understand, usually the solution is to point out you are not an expert in the topic of discussion and they need to explain the terms they are using, or elect for more common ones.However, some people use big words for a single reason: they think it makes them better than you, smarter than you, or more sophisticated than you. These people are easy to spot: they will use such vocabulary where other terms would actually be more appropriate (no, it wasn't serendipitous that the bus was on time, you mean it was fortunate) or they will slip up and use a term that is flat out not meant to be there. A key part of using advanced vocabulary just because that's what you use is, in fact, using it correctly. And when I do pick up that someone is attempting to use unusual, posh, or simply confusing language in an attempt to place themselves above me, I'll usually play them. A fun one is to repeatedly use words that sound posh but don't exist, and watch them either go along with it thinking you know words they don't, or realise what you're doing and struggle to decide if they can call you out on it.

A big part of the answer to this question depends on the place and society this 'every' person belongs to. However, I would try to give a more universal answer here:Driving and basic car upkeepManaging resources and budgeting; money, time, health, energyManaging people, creating and taking care of relationships; friends, family, colleagues etc., how to treat a lady wellLogic and critical thinkingBasic usage of computer and internet; how to use at least one OS, some necessary applications (word processor, spreadsheet etc.), how to Google  Everyday mathematicsThe fact that you can never be good enoughThe fact that hardwork is more important than intelligence and good fortuneHow to make independent/wise choices and prioritizingBasic survival skills; building a fire, cooking, basic first aid (CPR and Heimlich maneuver)When to fight and how to do itPacking lightRecognizing one's limits; alcohol/drug usage, frankness with people, independence of speech and action, when you are a guest somewhereBasic physical training; push-ups, sit-upsHow to play at least one sportGeography; names and locations of major countries, cities, physical features of the world and especially of the little bubble you live inHow to smileBehavioral control; when to promise, how to negotiate, how to be a good house guestTaking basic security precautions; physical and onlineTaking care of one's physical, mental, emotional health

To cover the analytic, no pun intended, side of things.Normally, early on in a complex analysis course the exponential function is defined in terms of the following power series:[math]\displaystyle e^z = \sum_{n=0}^{+\infty}\dfrac{z^n}{n!}, \; z\in\mathbb{C} \tag{1}[/math]which has an infinite radius of convergence because the following limit exists:[math]\displaystyle \lim_{n\to+\infty}\dfrac{a_n}{a_{n+1}} = \lim_{n\to+\infty}\dfrac{(n+1)!}{n!} = \infty \tag{2}[/math]which, in turn, tells us that the series in (1) converges for each complex number [math]z[/math] and that on every compact subset of [math]\mathbb{C}[/math] the convergence is uniform.Our two trigonometric functions in question are also defined as the following convergent power series:[math]\displaystyle \cos z = \sum_{n=0}^{+\infty}\dfrac{(-1)^nz^{2n}}{(2n)!}, \; z\in\mathbb{C} \tag{3}[/math][math]\displaystyle \sin z = \sum_{n=0}^{+\infty}\dfrac{(-1)^nz^{2n+1}}{(2n+1)!}, \; z\in\mathbb{C} \tag{4}[/math]Shortly after publishing his Principia in 1687 Newton was asked, in so many words, exactly how does the gravitational force work? What is its inner nature? Newton waved the question off with (in my words) don’t worry about it for now - I gave you a precise mathematical way to treat it, see how far you can take that.Similarly, the modern novice students, when given (1,3,4) may feel uncomfortable - where did these come from? Don’t worry about it for now. We gave you a precise mathematical way to treat them, see how far you can take it. Far enough.Since the series in (1) converges (uniformly) we can manipulate it as follows: let us first replace [math]z[/math] with [math]iz[/math] in it:[math]\displaystyle e^{iz} = \sum_{n=0}^{+\infty}\dfrac{i^nz^n}{n!} \tag{5}[/math]and then replace [math]z[/math] with [math]-iz[/math]:[math]\displaystyle e^{-iz} = \sum_{n=0}^{+\infty}\dfrac{(-1)^ni^nz^n}{n!} \tag{6}[/math]Because the series in (1) converges we can safely add (6) to (5):[math]\displaystyle e^{iz}+e^{-iz} = \sum_{n=0}^{+\infty}\dfrac{i^nz^n}{n!} + \sum_{n=0}^{+\infty}\dfrac{(-1)^ni^nz^n}{n!} \tag*{}[/math]and combine the terms, factoring out what is common:[math]\displaystyle e^{iz}+e^{-iz} = \sum_{n=0}^{+\infty}\dfrac{i^nz^n}{n!}\left(1 + (-1)^n\right) \tag{7}[/math]Notice that when [math]n[/math] is an odd natural number then the term [math]1 + (-1)^{2k+1} = 1 + (-1)^{2k}\cdot(-1)[/math] vanishes (for a suitable positive integer [math]k[/math]), thus taking all the corresponding terms in the series (7) into oblivion.We now simply re-number the dummy summation index [math]n[/math] in (7) so that while it runs through the set of natural numbers [math]\mathbb{N}[/math], only even integers participate in the game (we replace [math]n[/math] with [math]2n[/math]):[math]\displaystyle e^{iz}+e^{-iz} = \sum_{n=0}^{+\infty}\dfrac{i^{2n}z^{2n}}{(2n)!}\left(1+(-1)^{2n}\right) = \tag*{}[/math][math]\displaystyle 2\;\sum_{n=0}^{+\infty}\dfrac{i^{2n}z^{2n}}{(2n)!} \tag*{}[/math]By using the rules for the exponentiation of complex numbers, we have:[math]i^{2n} = \operatorname{exp}(2n\log i) = \operatorname{exp}\left(2n\log e^{\frac{i\pi}{2}}\right) = \tag*{}[/math][math]\operatorname{exp}\left(2n\cdot \dfrac{i\pi}{2}\right) = \operatorname{exp}(i\pi n) = (-1)^n \tag*{}[/math]If the above computation is too fancy then simply write[math]i^{2n} = \left(i^2\right)^n = (-1)^n \tag*{}[/math]which means that while an odd number of [math]i^2[/math] multiplies to a negative unity, an even number of [math]i^2[/math] multiplies to a unity.Either way we come to[math]\displaystyle e^{iz}+e^{-iz} = 2\;\sum_{n=0}^{+\infty}\dfrac{(-1)^nz^{2n}}{(2n)!} \tag{8}[/math]Juxtapose both sides of (8) against the same both sides of (3) to conclude that:[math]e^{iz}+e^{-iz} = 2\cos z \tag*{}[/math]from where[math]\cos z = \dfrac{e^{iz}+e^{-iz}}{2} \tag{9}[/math]Similarly, instead of adding (6) to (5) subtract (6) from (5):[math]\displaystyle e^{iz}-e^{-iz} = \sum_{n=0}^{+\infty}\dfrac{i^nz^n}{n!}\left(1 - (-1)^n\right) \tag{10}[/math]Now, however, when [math]n[/math] is an even natural number then the term [math]1 - (-1)^{2k}[/math] will vanish taking all the corresponding members of (10) with it. Thus, only the odd positive integers contribute to the result:[math]\displaystyle e^{iz}-e^{-iz} = \sum_{n=0}^{+\infty}\dfrac{i^{2n+1}z^{2n+1}}{(2n+1)!}\left(1 - (-1)^{2n}\cdot(-1)\right) = \tag*{}[/math][math]\displaystyle 2\;\sum_{n=0}^{+\infty}\dfrac{i^{2n+1}z^{2n+1}}{(2n+1)!}\tag*{}[/math]Again, you should be able to show that[math]i^{2n+1} = i^{2n}\cdot i = i\cdot(-1)^n \tag*{}[/math]bringing us to[math]\displaystyle e^{iz}-e^{-iz} = 2i\;\sum_{n=0}^{+\infty}\dfrac{(-1)^nz^{2n+1}}{(2n+1)!}\tag{11}[/math]Comparing (11) to (4), we conclude that[math]e^{iz}-e^{-iz} = 2i\sin z \tag*{}[/math]from where[math]\sin z = \dfrac{e^{iz}-e^{-iz}}{2i} \tag{12}[/math]Multiplying (9) through by [math]2[/math] and (12) by [math]2i[/math] and adding the results, we have:[math]2e^{iz} = 2\cos z + 2i\sin z \tag*{}[/math]or[math]e^{iz} = \cos z + i\sin z \tag{13}[/math]It may be tempting to think that due to a familiar shape the (13)’s RHS represents the real and the imaginary parts of (13)’s LHS but this is wrong - both [math]\cos z[/math] and [math]\sin z[/math] are complex numbers.Lastly, in (13) let [math]z[/math] be a real number [math]\theta[/math] (which it is perfectly allowed to be):[math]e^{i\theta} = \cos \theta + i\sin \theta \tag*{}[/math]which, hopefully, explains why and how the series representation of the (complex) exponential function leads to a relationship involving the (complex) trigonometric functions of cosine and sine.Further, by squaring both sides of (9) and adding the result to the square of (12) we find:[math]\cos^2 z + \sin^2 z = 1 \tag*{}[/math]Since all our series converge uniformly, their term by term differentiation is allowed. Simply move the differentiation operator through the summation sign:[math]\displaystyle \dfrac{\mathrm d}{\mathrm dz}e^z = \dfrac{\mathrm d}{\mathrm dz}\left(\sum_{n=0}^{+\infty}\dfrac{z^n}{n!}\right) = \sum_{n=0}^{+\infty}\dfrac{\mathrm d}{\mathrm dz}\left(\dfrac{z^n}{n!}\right) = \tag*{}[/math][math]\displaystyle \sum_{n=1}^{+\infty}\dfrac{nz^{n-1}}{n!} = \sum_{n=1}^{+\infty}\dfrac{z^{n-1}}{(n-1)!} = \tag*{}[/math](by changing the dummy summation variable from [math]n-1[/math] back to [math]n[/math]. If that is too confusing at first then work formally by introducing the [math]n - 1 = k[/math] substitution and observing that while [math]n[/math] varies starting from [math]1[/math], [math]k[/math] varies starting from [math]0[/math]. When done - replace [math]k[/math] with [math]n[/math])[math]\displaystyle \sum_{n=0}^{+\infty}\dfrac{z^n}{n!} = e^z \tag*{}[/math]and we come to a property of the exponential function which can also be defining:[math]\dfrac{\mathrm d}{\mathrm dz}e^z = e^z \tag*{}[/math]As a good exercise, do prove on your own that[math]\dfrac{\mathrm d}{\mathrm dz}\cos z = - \sin z \tag*{}[/math][math]\dfrac{\mathrm d}{\mathrm dz}\sin z = \cos z \tag*{}[/math]and so on.