Integrate Mark Test with airSlate SignNow

Integrate mark test

Professor Dave again, let’s check out some tests. We just learned about convergent series, which are ones where the sum of all the terms in the series is some finite number. But it can often be difficult to calculate the precise value of the sum, and there are only certain special cases where we can easily get the sum using simple computation. Luckily there are special tricks we can use to help us find sums, or at least estimate them, and one such trick is called the integral test. Take something like the sum of one over N squared, from one to infinity, giving us one over one squared, plus one over two squared, plus one over three squared, and so forth, and we can easily plot these points graphically. If we also show the function Y equals one over X squared, essentially connecting the dots, we recognize that this is the precise example we looked at when we learned about improper integrals, and we know the area under the curve over this infinite interval is actually finite. Therefore, we shouldn’t be surprised to learn that this similar looking series is convergent, especially since the sequence one over N squared does go to zero as N approaches infinity. In fact, we can actually use this function to estimate the sum of the series. Let’s make some rectangles under the curve, each with a base of one and a height equal to the value of the function at the right endpoint of the interval. The area of each rectangle is equal to one of the terms in the series, because this rectangle has an area of one over two squared, and the second term in the series is one over two squared. This rectangle has an area of one over three squared, and the third term in the series is one over three squared. So every rectangle all the way to infinity has a corresponding term in the series, and we are leaving out only the first term, one. The sum of the areas of these rectangles is pretty close to being the area under the curve, which we know is just the integral of the function. So the sum of the series will be approximately equal to one, to account for the first term in the series that has no corresponding rectangle, plus this improper integral, which approximates the sum of all the rectangle areas and therefore accounts for the rest of the terms in the series. We know how to evaluate the integral, we just find the antiderivative, which is negative one over X, and we evaluate at infinity and then at one, which gives us zero minus negative one, or positive one. The one from before plus the one from the integral equals two, so that’s our estimate for this sum. Now we know that the rectangles don’t quite fill up the total area, as there are some empty spaces, so the integral is bigger than what we...