Question

Find whether the following series are convergent or divergent:
$1+\frac{1}{2}x+\frac{|\underset{–}{2}}{3^2}{x}^2+\frac{|\underset{–}{3}}{4^3}{x}^3+\frac{|\underset{–}{4}}{5^4}{x}^4+....$

Answer 1

Let the general term for series be $T_n$

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

Applying ratio test

$\underset{n\to \infty }{lim}\frac{T_{n+1}}{T_n}=\underset{n\to \infty }{lim}\frac{n!x^n}{(n-1)!x^{n-1}}{\left(\frac{n}{n+1}\right)}^n\underset{n\to \infty }{lim}\frac{n}{x}{\left(\frac{1}{1+\frac{1}{n}}\right)}^{\infty }=\infty >1$

The given series is divergent.