Question

Determine whether the series converges or diverges...

Answer 1

Convergence

Explanation:

Use the Squeeze Theorem,

I know that

`\frac{k \sin {}^(2) (k) }{1 + k {}^(3) }`

lies between 0 and 1 so

`0 \leqslant \frac{k \sin {}^(2) (k) }{1 + {k}^(3) } \leqslant \frac{k}{1 + k {}^(3) }`

The final series behaves like

`\frac{1}{k {}^(2) }`

Using the p series, since p is 2, the series

`\frac{k}{1 + k {}^(3) } \: converges`

Since the 0 and k/1+k^3 converges, the series converge.

Convergence