Question

Determine whether the series is convergent or divergent.

We use the Integral Test here which is:

• If \(\int_1^\infty f(x) \, dx\) convergent, then \(\sum_{n=1}^{\infty} a_n\) is convergent.

• If \(\int_1^\infty f(x) \, dx\) convergent, then \(\sum_{n=1}^{\infty} a_n\) is divergent.

Then you basically take the \(\lim_{m \to \infty} \int_1^m f(x) \, dx\) which in this case is \(\frac{1}{n^{\sqrt{2}}}\).

You proceed to take the integral with Power Rule and then finally taking the limit.

Since the limit exists, it converges which is the final answer.

The power-P rule of the Integral Test can also be applied here!

Solutions

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