Question

Two series 2an and 2for which an bn and bn converges but 2an 1-1 does not.

This is all of the information I was given

The question is simply to come up with an example of when this would happen by giving an “an” and “bn”

Answer 1

Given two series $\sum_{n=1}^{\infty }a_n$ and $\sum_{n=1}^{\infty }b_n$ . Also . Then by comparison test, if $\sum_{n=1}^{\infty }b_n$ converges then $\sum_{n=1}^{\infty }a_n$ must converge. So the answer is $\sum_{n=1}^{\infty }a_n$ converges.

Example:

$a_n=\frac{1}{n^2+4},b_n=\frac{1}{n^2}$

Since the p-series $\sum_{n=1}^{\infty }\frac{1}{n^2}$ is convergent by comparison test , the series $\sum_{n=1}^{\infty }\frac{1}{n^2+4}$ converges.