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JH
Numerade Educator

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Problem 27 Easy Difficulty

Determine whether the series converges or diverges.
$ \displaystyle \sum_{n = 1}^{\infty} \left( 1 + \frac {1}{n} \right)^2 e^{-n} $

Answer

Converges

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Video Transcript

to determine whether the Siri's convergence or diverges now for this one. Let's use limit comparison from this section eleven point four and in your textbook. Now, if I call this and let me call being to be won over either, then now we look at a n overby end in the limit so that'LL just be after canceling out the ease and this is just one now. This is important because if we want to use little comparison, we must have that the limit is in between zero and infinity. If the limit exists. Of course, in our case, that exist and equals the one so we can apply the test now, since to some of the bm thiss converges on what we can actually show, this I can write. This is the sum of one of the e to the end, and this conversion is because this is a geometric with our equals, one over easy, and that satisfies the an apology that it's less than one and absolute value, and that's what gives us the conversions. Now we apply the test by Lim comparison our Siri's Gin. There's our cinema, our Siri's of one plus one over end square. You know, the minus n also converges help, and that's our final answer