Convergency Test

Suppose $x_n \ge 0$, $\sum_{n=1}^{\infty} \frac{x_n}{x_n+1}$ converges if and only if $\sum_{n=1}^{\infty} x_n$ converges.

Solution:

As $$
0 \le \frac{x_n}{1+x_n}\le x_n,\\
\sum {x_n}<\infty\Rightarrow \sum \frac{x_n}{1+x_n}<\infty
$$


Now suppose that $\sum \frac{x_n}{1+x_n}<\infty $.

$\frac{x_n}{1+x_n}\to 0$, and so $x_n\to 0$, so there is a $N$ such as
$
n>N\Rightarrow x_n<\frac 12
$
and then$$
\frac{x_n}{1+x_n}>\frac 23 x_n
$$so $$
\sum \frac{x_n}{1+x_n}<\infty \Rightarrow \sum {x_n}<\infty
$$