Problem 18 Easy Difficulty

Determine whether the series is convergent or divergent.
$ \displaystyle \sum_{n = 1}^{\infty} \frac {1}{n^2 + 2n + 2} $

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Clayton C.

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So we want to determine if this Siri's is going to be convergent or divergent. Um, And to do this, we're going to use the integral. So since we have this summation from one to infinity, we're going to take the integral from one to infinity, and it's going to be integral of one over e squared plus two X plus two Onda. We can also look at that as taking the limit as be approaches infinity from one to be. But doing this, we can use substitution method. Andi, ultimately, they're using substitution method. What we get is that for the integrate this expected X I don't answer is pi over two minus the arc tangent. Okay, we see that we get the same answer. So more importantly, though, we see that this answer does not go to infinity. Rather, it is a finite value. So with that being considered, we know that this Siri's must converge. Andi, that's just by determining how we set up this problem and then evaluating the integral Um, using these calculators is extremely quick and useful. But even if you're doing it by hand, we just recognize how we solve intervals and we see that we'll end up getting the same answer