$\"\"$

\

Consider $\"\"$ and $\"\"$ .

\

Direct comparision test :

\

If $\"\"$ for all $\"\"$ .

\

1. If $\"\"$ converges, then $\"\"$ converges.

\

2. If $\"\"$ diverges, then $\"\"$ diverges.

\

By comparision test, prove that $\"\"$ .

\

Find the value of $\"\"$ so that $\"\"$ .

\

Consider function $\"\"$ .

\

$\"\"$ , when $\"\"$ .

\

$\"\"$ for $\"\"$ .

\

Therefore, $\"\"$ , $\"\"$ .

\

$\"\"$ for $\"\"$ .

\

$\"\"$ for $\"\"$ .

\

Therefore, $\"\"$ .

\

$\"\"$

\

Consider $\"\"$ .

\

The series $\"\"$ is converges by $\"\"$ -series test.

\

Since $\"\"$ is converges, by comparision test $\"\"$ is converges.

\

$\"\"$

\

The statement is proved.