Question

Use the Integral Test to 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 \(\frac{n^2}{n^3+1}\).

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

Since the limit does not equal to 0, it diverges which is the final answer.

Solutions

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