Question

Find whether the following series are convergent or divergent:
$1+a+\frac{a(a+1)}{1.2}+\frac{a(a+1)(a+2)}{\mathrm{1.2.3}}+...$

Answer 1

Let $T_n$ be the general term of the series

$T_n=\frac{a(a+1)(a+2).....[a+(n-1\left)\right]}{n!}{T}_{n+1}=\frac{a(a+1)(a+2).....(a+(n-1\left)\right)(a+n)}{(n+1)!}$

Applying ratio test

$\underset{n\to \infty }{lim}\frac{T_{n+1}}{T_n}=\frac{(a+n)n!}{(n+1)!}=\underset{n\to \infty }{lim}\frac{(a+n)}{(n+1)}=1$

Ratio test fails.

We know apply Raabe's test

$\underset{n\to \infty }{lim}n[\frac{T_n}{T_{n+1}}-1]=\underset{n\to \infty }{lim}n[\frac{1+n}{a+n}-1]=\underset{n\to \infty }{lim}n\frac{[1+n-a-n]}{a+n}=\frac{n(1-a)}{n(1+\frac{a}{n})}=(1-a)$

If $1-a>1a<0$

The series is convergent

If $1-a<1a>0$

The series is divergent