Question

Determine if the series is convergent or divergent using the comparison test.

Answer 1

Clearly,
`\frac1{k√(k+2)}<\frac1{k\sqrt k}=\frac1{k^(3/2)}` for all
`k\ge1`.

Recall that


`\displaystyle\sum_(k\ge1)\frac1{k^p}=\begin{cases}\text{converges}&\text{for }p>1\\\text{diverges}&\text{otherwise}\end{cases}`

Since
`p=\frac32>1`, it follows that


`\displaystyle\sum_(k\ge1)\frac1{k\sqrt k}`

also converges, which in turn means that the series


`\displaystyle\sum_(k\ge1)\frac1{k√(k+2)}`

does too.