Question

Why does this series converge?
∞
∑ (n+1)/[(n)(n+2)(n+3)]
n=1

Answer 1

Compare the series to the convergent series,

$\displaystyle\sum_{n=1}^\infty\frac1{n^2}$

By the limit comparison test, the given series converges because

$\displaystyle\lim_{n\to\infty}\frac{\frac{n+1}{n(n+2)(n+3)}}{\frac1{n^2}}=\lim_{n\to\infty}\frac{n(n+1)}{(n+2)(n+3)}=1$