Answer to Question #188122 in Real Analysis for Nikhil

Question

Give an example of an series Σan such that Σan is not convergent but the sequence (an) converges to 0

Answer 1

Solution:

Assume $a_n=\dfrac{\ln (n)}{n}$

$\lim _{n \rightarrow \infty} \frac{\ln n}{n}$

$\\=\lim _{n \rightarrow \infty} \frac{1 / n}{1}$ [Using L' Hopital rule]

$=0$

Next, $\Sigma a_n= \sum _{n=1}^{\infty \:}\frac{\ln \left(n\right)}{n}$

Consider $\int_{1}^{\infty} a_{n} d n=\int_{1}^{\infty} \frac{\ln n}{n} d n=\left[\frac{(\ln n)^{2}}{2}\right]_{1}^{\infty}= diverges$

So, by integral series test, $\int_{1}^{\infty} a_{n} d n$ is divergent, then so is $\sum_{n=1}^{\infty} a_{n}$

Thus, we have $\sum_{n=1}^{\infty} \frac{\ln n}{n}$ is divergent.