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{(n+1)/(n(n+2)(n+3))}{\frac1{n^2}}=\lim_(n\to\infty)(n(n+1))/((n+2)(n+3))=1`