$$\lim\limits_{n \to \infty} \frac 1{n\log \left ( 1+\frac1n \right ) }$$
I start by writing $\lim\limits_{n \to \infty} \frac 1n=0$, and $\lim\limits_{n \to \infty} \frac 1{n+\log \left ( 1\frac1n \right ) }=0$
Since limit exists, by multiplication law, $\lim\limits_{n \to \infty} \frac 1{n\log \left ( 1+\frac1n \right ) }=0$
Hence the series $ \sum_{n}^{\infty }\frac{1}{n\log 1+\frac{1}{n}}$ converge.
However, the $\lim n \to \infty \frac{1}{n\log \left ( 1+\frac{1}{n} \right ) }$ is given to be 1 in the solution, would really like to know where I've done wrong.
Thanks for the help.
Answer
Your first limit $$\lim_{n\to\infty}\frac{1}{n+\log(1+\frac{1}{n})}=0$$
In the corrected case we have $$\lim_{n\to \infty}\frac{1}{\log\left(1+\frac{1}{n}\right)^n}=1$$ since $$\lim_{n\to \infty}\log\left(1+\frac{1}{n}\right)^n=\log(e)=1$$ if $\log$ means logarithmus naturalis, to the base $e$
No comments:
Post a Comment