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Numerade Educator

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Problem 24 Medium Difficulty

Determine whether the series is convergent or divergent.
$ \displaystyle \sum_{k = 1}^{\infty} ke^{-k^2} $

Answer

convergent

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Video Transcript

for this Siri's. We shall use the comparison test So the Siri's we can rewrite as from K equals one infinity of okay e to the negative k to the K. And what's important is that either the negative k to the K you know, really would be on the denominator. Sophie, write it that way. It might make it clear what we're about to do, since either the K to the K is always going to be bigger than just each of the k. No turning that off to the side either. The K the K is greater than either the k mom since Kei is greater than one or equal to. So that means that we end up with something smaller than Siri's of just K over you do the K or rewriting it back. Okay, e to the negative k power and this Siri's we know converges Bye ratio test. So if we have they an lesson or equal to be in and being converges, and in this case being you're there are Anne was our original Siri's, we see that are the serious that we want is always smaller than another conversion. Siri's So we have conversion. Bye compares