Problem 32 Medium Difficulty

Determine whether the series converges or diverges.
$ \displaystyle \sum_{n = 1}^{\infty} \frac {1}{n^{1 + 1/n}} $

Answer

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Video Transcript

Let's determine whether is the Siri's convergence or diverges. So here I'LL use limit comparison Test two. Let's call this term over here am I? And then I'll call Being as then gets really, really big this one over ends really small. So this is called being to be one one over into the one power. So we're ignoring the one over and on the A and Salama comparison says we look at the limit is N goes to infinity of an over bian. That's just the limit as N goes to infinity. So dividing by one over end is the same thing is multiplying by N. And then I could cancel out one of these ends. We just have the limit as n goes to infinity of one over and to the one over end, and this limit will be one. And you can kind of show this here using Let's replace and with X. And let's just look at X to the one over X. So here I can rewrite this as e to the natural Log X is one over X. This is just using the fact that e and natural log of any number is just equal to that number because Ian Ellen X are in verses. And then here I can use a lot of property toe. Pull this one over X outside of the log. And then as X goes to infinity, we have that Ellen X goes to infinity. So we have Ellen next that goes to infinity. But then we have excellent denominated these air, both going to infinity. So we should use Low Patel's rule here. So we take the derivative of the top natural lot. That's just one over X, and then we take the derivative of the denominator X. That's just one. So we have e one over infinity that's equal to eat to the zero, which is equal to one now. We could use limit comparison test because the limit is a number that's bigger than zero and less than infinity. This is what's in the condition that we must satisfy. If we want to use limit comparison, let me take a step back. We just showed that this thing goes toe one so and that shows that this denominator those toe one in the limit So we just have one over one equals one. That's the That's where we're using women comparison. So since there's some diverges, we know this one diverges. You could either use the pee test or just a harmonic series. Our Siri's We'LL also diverge by living comparison, and that's our final answer.