Question

Determine whether the series is absolutely convergent, conditionally convergent,or divergent.

\(\sum\limits_{n = 1}^\infty {\frac{{n!}}{{{{100}^n}}}} \)

Short answer

The Series is Absolutely Convergent.

Step-by-step solution

(Using Ratio Test)

\({\lim _{n \to \infty }}|\frac{{{a_{n + 1}}}}{{{a_n}}}|\)=L

Then Series is Absolutely Convergent if\(L < 1\),

And Divergent When\(L > 1\)or\(\infty \)\(\)

(Check For Convergence..)

\({\lim _{n \to \infty }}|\frac{{{a_{n + 1}}}}{{{a_n}}}|\)\( = {\lim _{n \to \infty }}|\frac{{\frac{{(n + 1)!}}{{{{100}^{n + 1}}}}}}{{\frac{{n!}}{{{{100}^n}}}}}|\)

=\({\lim _{n \to \infty }}|\frac{{(n + 1)*n!}}{{{{100}^n}.100}}*\frac{{{{100}^n}}}{{n!}}|\)

=\(|\frac{{(n + 1)}}{{100}}|\)

\( = \infty \)

=\(\infty > 1\)

Hence,By Ratio Test The Series is Divergent.

\(\)..