Tuesday, April 10, 2018

real analysis - Does the series $sumlimits_{n=1}^{infty }left ( 1-frac{ln(n)}{n} right )^{2n}$ diverge?



could anyone help me figure out whether this infinite series
$$\sum_{n=1}^{\infty }\left ( 1-\frac{\ln(n)}{n} \right )^{2n}$$
diverges?



I've tried using Cauchy's and d'Alembert's limit tests but both gave the result 1. I've also tried the necessary condition for convergence, but
$$\lim_{n\rightarrow \infty }\left ( 1-\frac{\ln(n)}{n} \right )^{2n}=0$$


Answer




Use



$$\sum_{n=1}^\infty \Bigl( 1 - \frac{\log n}{n} \Bigr)^{2n} \leq \sum_{n=1}^\infty \biggl(\exp\Bigl( -\frac{\log n}{n} \Bigr)\biggr)^{2n} =\sum_{n=1}^\infty \exp( - 2\log n ) = \sum_{n=1}^\infty \frac{1}{n^2} < \infty.$$


No comments:

Post a Comment

analysis - Injection, making bijection

I have injection $f \colon A \rightarrow B$ and I want to get bijection. Can I just resting codomain to $f(A)$? I know that every function i...