$\"\"$

\

The binomial series is $\"\"$ .

\

$\"\"$ can be written as $\"\"$ .

\

Binomial series is $\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$ .

\

$\"\"$

\

Now find the radius of convergence,using the ratio test

\

\

The ratio test states that

\

(1).If $\"\"$ then the series converges absolutely.

\

(2).If $\"\"$ then the series does not converge.

\

(3).If $\"\"$ or the limit fails to exist then the test is inconclusive.

\

\

$\"\"$

\

\"\"

\

Therefore the result is same, converges if

\"\"

, therefore

\"\"

.

\

\

$\"\"$

\

\

\"\"

and

\"\"

.

\

\