Question

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

\(\sum\limits_{n = 1}^\infty {{{( - 1)}^{n - 1}}\frac{n}{{{n^2} + 4}}} \)

Short answer

The Series is Conditionally Convergent.

Step-by-step solution

(Using Leibnitz Test)

\({U_n} > {U_{n + 1}}\)

But in this question\({u_1} < {u_2}\)

Therefore Leibnitz test fails The series can’t be Absolutely Convergent..

(Check For Convergence..)

\({\lim _{n \to \infty }}{b_n} = {\lim _{n \to \infty }}(\frac{n}{{{n^2} + 4}})\)\( = {\lim _{n \to \infty }}\{ \frac{1}{{n(1 + \frac{4}{{{n^2}}})}}\)=0

Hence,The Series Satisfy two Conditions for alternating Series Test..

\(\)Therefore,The Series is Conditionally Convergent..