Question

Use the ratio test to show that a binomial series converges for$\left|x\right|<1$

Short answer

The statement has been proven

Step-by-step solution

Given Information 

The binomial series.

Definition of the binomial series

The Taylor series for the function given by is the binomial series, where is an arbitrary complex number.

Prove the statement.

The binomial series states that$(1+x{)}^p=\sum _{n=0}^{\infty }\left(\begin{array}{l}p\\ n\end{array}\right)x^n$

$\left(\begin{array}{l}p\\ n\end{array}\right)=\frac{p!}{n!(p-n)!}(1+x{)}^p$

$=\sum _{n=0}^{\infty }\frac{p!}{n!(p-n)!}{X}^n{a}_n$

$=\frac{p!}{n!(p-n)!}{x}^n{a}_{n+1}$

$=\frac{p!}{(n+1)!(p-n-1)!}{x}^{n+1}$

$Let{\rho }_n=\left|\frac{a_{n+1}}{a_n}\right|\text{,}$

Substitute the value of the power series in the formula above, the equation becomes as follows,

${\rho }_n=\left|\frac{a_{n+1}}{a_n}\right|$

$=\left|\frac{\frac{p!}{(n+1)!(p-n-1)!}{x}^{n+1}}{\frac{p!}{n!n}{x}^n}\right|$

$=\left|\frac{p!n!x^{n+1}(p-n)!}{(n+1)!(p-n-1)!p!x^n}\right|$