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Problem 9 Medium Difficulty

Test the series for convergence or divergence.

$ \displaystyle \sum_{n = 0}^{\infty} ( - 1)^n \frac {\pi^{2n}}{(2n)!} $

Answer

Converges

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Video Transcript

for this problem, we can use the ratio test. So these terms here going to be our A in terms for the ratio test. What we're looking is to see if limit as in approaches infinity of the absolute value of hey seven plus one over Mason. And we're curious if that's less than one. Okay, so when we look at this limit, the absolute value signs is just going to tell us that we can throw away this minus one to the end. And then we're gonna have hi to the two times and plus one divided by two times and plus one that whole thing factorial. And then we're going to be dividing by the in terms, which is the same as multiplying by the reciprocal. So we'LL be multiplying by two in factorial over pi to the two n lips. A little bit of difficulty here. Okay, so we're curious to see if that's strictly last on one. If it happens to be equal to one, then it just means that the test is inconclusive. It could be convergence. It could be divergence. But if it's less than one, they would know that these terms are approaching zero fast enough to where we get convergence toe happen Kiss when work this out. So this is pi to the two end plus two after we expand this. So we get the two ends to cancel out here. So after we make some cancellations with their pie terms, we just get pie squared can if we remember how the factorial works. Five factorial, for example, is five times four times, three times, two times one. So too in factorial is one times two times three times for all the way up toe times two in. And then down here. We're doing the same type of multiplication, except we're going all the way up toe two and plus two. After we expand that, there's going to be a ton of cancellations that are going to be occurring. Everything is going to cancel out with these guys, except for the last two terms in the product that we're looking at here because the last two terms being two in plus one and to end plus two. Okay, these guys are not going to be cancelled out with anything from over here, but everything else will be cancelled out if you just remember how the factorial function works. Okay, so this limit turns out to be zero. So it's We're definitely going to end up getting convergence. That limit didn't have to be zero to get convergence here. We just want it to be something less than one. So that zero, then we're we're definitely good. So this is going to be a convergent Siri's.