Question

Determine whether the series is convergent or divergent.

First of all, we try to find out the summation series of the series.

Then, 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 \(\frac{1}{n^2 + n^3}\).

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.

Solutions

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