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Problem 19 Easy Difficulty

Test the series for convergence or divergence.

$ \displaystyle \sum_{k = 1}^{\infty} ( - 1)^n \frac {\ln n}{\sqrt {n}} $

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Converges

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Video Transcript

for this problem. We want to determine if the Siri's is going to converge or diverge on. There's a lot of different tests that we can use this, but in this case, the best one would be an alternating serious test. So with the alternating serious test, what we can do is we can, um, take the derivative of the natural log of X over the square root of X. Um and we see that, but we'll end up getting is that the derivative will be negative for X greater than e squared eso. Now we can take the limit of the natural log of n over the square root of N. I mean, that's the limit as an approaches infinity. So then, using look, he tells rule, we can evaluate this eso we see that one. This is gonna be one over n and down here would be one over to fruit, and I'm going to be equal to okay the limit of to square root of n over and just just, um, one could be zero. That's the limit eso by the alternating Siri's tests. Since that zero, and since the derivative is negative, that's decreasing for large enough value of n We see that by the alternating Siri's test, Um, the given serious that we have will converge.