Integrate Signed Test with airSlate SignNow

Integrate signed test

consider the series from 1 to infinity of 1 over n plus 2 squared so will this series converge or diverge so in this video we're going to use the integral test to find the answer so let's say the sequence a sub n can be described as a function of n in order for the integral test to work the function must be positive it has to be continuous and it has to be decreasing now all of this must be true when x is equal to a greater than one so it has to be true on the interval from one to infinity so if these three conditions are satisfied then we can use the integral tests so here's the basic idea behind the integral test if you take the integral from one to infinity of f of x dx and let's say you get a finite number which we'll call n then the integral converges which means that the series will also be convergent now let's say if you take the integral but you don't get a finite number let's say if you get infinity or negative infinity then the integral diverges and so if the integral diverges according to the integral test the series must also diverge and so that's the basic idea behind the integral test so now let's finish working on this problem so let's write the corresponding function f of x which is one over x plus two squared so is this function positive would you say we're looking at x plus two squared regardless of what value that we plug in it's always positive except when x is negative two because we're going to have a vertical asymptote there so for the most part this is a positive function now is it continuous because we have a vertical asymptote at negative two that's basically an infinite discontinuity at that point so it's not continuous everywhere so does it satisfy this condition now remember it only has to be continuous on this interval and not everywhere so is this discontinuity is it in the interval negative two is not in the interval so from one to infinity it's continuous everywhere in this interval therefore the second criteria has been satisfied now let's focus on the third point is the function always decreasing the way we can determine that is by taking the first derivative or using the first derivative test if the first derivative is negative then the function is decreasing so we need to show that the first derivative is negative on this interval so let's go ahead and find the first derivative so let's begin by rewriting it as x plus 2 raised to the minus 2. and now let's take the first derivative of that function so using the power rule it's going to be negative 2 times x plus 2 and then negative two minus one is negative three and then you need to multiply by the derivative of...