Question

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

\(\sum\limits_{n = 0}^\infty {\frac{{{{( - 3)}^n}}}{{(2n + 1)!}}} \)

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..)

\({\lim _{n \to \infty }}|\frac{{{a_{n + 1}}}}{{{a_n}}}|\)\( = {\lim _{n \to \infty }}|\frac{{\frac{{{{( - 3)}^{n + 1}}}}{{(2n + 3)!}}}}{{\frac{{{{( - 3)}^n}}}{{(2n + 1)!}}}}|\)

=\({\lim _{n \to \infty }}|\frac{{( - 3)}}{{(2n + 3)(2n + 2)}}|\)

=\(\begin{aligned}\frac{{ - 3}}{\infty } &= 0\\\end{aligned}\)

=\(0 < 1\)

Hence,By Ratio Test The Series is Convergent..

\(\)Therefore From the Results,The given Series is Absolutely Convergent..