Question

Show that the series
$1+\frac{2^p}{|\underset{–}{2}}+\frac{3^p}{|\underset{–}{3}}+\frac{4^p}{|\underset{–}{4}}+....$
is convergent for all values of $p$.

Answer 1

To show that the series $1+\frac{2^p}{2!}+\frac{3^p}{3!}+\frac{4^p}{4!}+...$

is convergent for all values of $p$.

Here,

$a_n=\frac{n^p}{n!}$

We are using series ratio test

If there exists an $N$ so that for all $n\ge N$, $a_n\ne 0$

and $L=\underset{n\to \infty }{lim}\left|\frac{a_{n+1}}{a_n}\right|$

1 ) If $L<1$, then $\sum a_n$ converges

2) If $L>1$, then $\sum a_n$ diverges

3) If $L=1$, then the ratio test is inconclusive

$\underset{n\to \infty }{lim}\left|\frac{\frac{(n+1{)}^p}{(n+1)!}}{\frac{n^p}{n!}}\right|$

$\underset{n\to \infty }{lim}\left|\frac{(n+1{)}^p}{(n+1)n^p}\right|$

$\underset{n\to \infty }{lim}\left|\frac{(n+1{)}^{p-1}}{n^p}\right|$

On applying limit, we get

$L={lim}_{n\to \infty }\left|\frac{(n+1{)}^{p-1}}{n^p}\right|$

$L=0$

Since, $L<1$

Hence, the given series converges.