## 17Calculus Infinite Series - Practice Problem 2442

Infinite Series Practice Problem 2442

$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{n^{3/2}+1} } }$$

Determine the convergence or divergence of the series.
If the series converges,
- determine the value the series converges to, if possible; and
- determine if the series converges absolutely or conditionally.

Choosing Which Tests To Try

Rejecting the Obvious

1. We can tell right away that this is not an alternating series, telescoping series or one of the other special series. So we will not be able to determine what the series converges to.
2. We can tell that the limit of $$a_n$$ is zero. So the divergence test will not help us.
3. The integral is not easily evaluated, so we will come back to it if none of the other tests work.

What We Are Left With

4. We are left with the ratio test, the comparison tests and the root test.

Applying The Tests

### Root Test

Test Summary List and Answer

test/series works? notes no limit is zero, so test is inconclusive no not a p-series no not a geometric series no not an alternating series no not a telescoping series no inconclusive yes yes no too complex no inconclusive

Final Answer

The series$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{n^{3/2}+1} } }$$ converges.

Really UNDERSTAND Calculus

### Topics You Need To Understand For This Page

 all infinite series topics L'Hopitals Rule

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