Question

Test the following series for convergence.

5.$\sum _{n=2}^{\infty }\frac{{(-1)}^n}{\ln n}$

Short answer

The series$\sum _{n=2}^{\infty }\frac{{(-1)}^n}{\ln n}$ converges.

Step-by-step solution

Given information

It is given that the series is $\sum _{n=2}^{\infty }\frac{{(-1)}^n}{\ln n}$.

Convergence Test

According to the convergence test for alternating series: An alternating series converges if the absolute value of the terms decreases steadily to zero, that is, if $\left|a_{n+1}\right|\le \left|a_n\right|$, and$\underset{n\to \infty }{\lim }{a}_n=0$.

Apply properties of Convergence Test

Let $a_n=\frac{{(-1)}^n}{\ln n}$.

Then, $a_{n+1}=\frac{{(-1)}^{n+1}}{\ln (n+1)}$.

Apply the convergence test as $\left|a_{n+1}\right|\le \left|a_n\right|$.

$\begin{array}{c}\left|\frac{{(-1)}^{n+1}}{\ln (n+1)}\right|\le \left|\frac{{(-1)}^n}{\ln n}\right|\\ \frac{1}{\ln (n+1)}<\frac{1}{\ln n}\end{array}$

Since the denominator is bigger, therefore the overall term is lesser, so the first property is satisfied.

Also, for $\underset{n\to \infty }{\lim }{a}_n=0$:

$\begin{array}{l}\underset{n\to \infty }{\lim }\frac{{(-1)}^n}{\ln n}=0\\ \underset{n\to \infty }{\lim }\frac{{(-1)}^{\infty }}{\ln \left(\infty \right)}\approx 0\end{array}$

Since the denominator is approaching to infinity, so the overall term will approach to zero.

The second property is also satisfied.

Therefore, it is proved that the series converges.