Question

Determine whether the following series converge:
∑_(k=1)^∞〖(-1)〗^(k+1) (k+3)/(k(k+1))

Answer 1

$\sum\limits_{k = 1}^\infty {\frac{{{{( - 1)}^{k + 1}}(k + 3)}}{{k(k + 1)}}}$

$\begin{array}{l} 1)\mathop {\lim }\limits_{k \to \infty } |{a_k}| = \mathop {\lim }\limits_{k \to \infty } \frac{{k + 3}}{{k(k + 1)}} = 0\\ 2)|{a_k}| - |{a_{k + 1}}| = \frac{{k + 3}}{{k(k + 1)}} - \frac{{k + 4}}{{(k + 1)(k + 2)}} = \frac{{(k + 3)(k + 2) - (k + 4)k}}{{k(k + 1)(k + 2)}} = \\ = \frac{{{k^2} + 5k + 6 - {k^2} - 4k}}{{k(k + 1)(k + 2)}} = \frac{{k + 6}}{{k(k + 1)(k + 2)}} > 0 \Rightarrow |{a_k}| > |{a_{k + 1}}| \end{array}$

Then, by Leibniz criterion, this series converges