Question

Use root test to determine the following series is convergent/divergent or inconclusive
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Answer 1

By the root test, the series converges. We have

`\displaystyle\lim_(k\to\infty)\sqrt[k]\left=\lim_(k\to\infty)\frac{\sqrt[k]{k^3}}3=\frac13\lim_(k\to\infty)k^(\frac3k)=\frac13`

which is less than 1.

In case it's not clear why
`k^(3/k)` should converge to 1:

`\displaystyle\lim_(k\to\infty)k^(\frac3k)=\lim_(k\to\infty)\exp\left(\ln k^(\frac3k)\right)=\exp\left(\lim_(k\to\infty)\frac{3\ln k}k\right)\to e^0=1`