$\"\"$

\

The binomial series is $\"\"$ .

\

Rewrite in the form of $\"\"$ .

\

$\"\"$ .

\

Binomial series is $\"\"$

\

$\"\"$ .

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

Represent the largest $\"\"$ numbers in the numerator by $\"\"$ .

\

Multiply the first $\"\"$ terms by $\"\"$ to obtain a simple general formula.

\

$\"\"$

\

$\"\"$

\

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 series converges if

\"\"

or

\"\"

or

\"\"

, then

\"\"

.

\

$\"\"$

\

\

\"\"

,

\"\"

.