Question

Determine whether the series is convergent or divergent. If its convergent, find its sum.

\(\sum\limits_{k = 1}^\infty {\frac{{k(k + 2)}}{{{{(k + 3)}^2}}}} \)

Short answer

The given geometric series isdivergent.

Step-by-step solution

Test for divergence.

If \(\mathop {\lim }\limits_{n \to \infty } {a_n}\)does not exist or if \(\mathop {\lim }\limits_{n \to \infty } {a_n} \ne 0\), then the series \(\sum\limits_{n = 1}^\infty {{a_n}} \)is divergent.

Using limits.

\(\mathop {\lim }\limits_{n \to \infty } {a_n} = \mathop {\lim }\limits_{k \to \infty } \frac{{k(k + 2)}}{{{{(k + 3)}^2}}} = \frac{{1(1 + 2)}}{{{{(4)}^2}}} \ne 0\)

So, the series diverges by the test of divergence.

Hence, \(\sum\limits_{k = 1}^\infty {\frac{{k(k + 2)}}{{{{(k + 3)}^2}}}} \) is divergent.