$\"\"$

\

The binomial series is $\"\"$ .

\

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

\

Binomial series is $\"\"$

\

Therefore substiute $\"\"$ and $\"\"$ .

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

If we move the starting from $\"\"$ to $\"\"$ then

\

$\"\"$ .

\

$\"\"$

\

Now find the radius of convergence, the series is not much different than the standard binomial series form so,

\

probably this will converge when $\"\"$ or $\"\"$ .

\

\

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 inclusive because there exist both

\

convergent and divergent series that satisfy this case.

\

\

\"\"

\

\

\"\"

\

\

Therefore the result is same, converges if

\"\"

, therefore

\"\"

.

\

$\"\"$

\

\

\"\"

and

\"\"

.