Show that the series is convergent.


Show that the series is convergent.

{eq}\Sigma_{n = 0}^{\infty} \frac{1}{1 + n^2} {/eq}

Convergence/Divergence for a Power Series:

There are many tests to determine whether the series is convergent or divergent. In this problem, we will use the comparison test and p-series test.

The p-series test states that:

If the series is of the form {eq}\displaystyle \sum_{n=1}^{\infty} \frac{1}{n^{p}} {/eq}

If {eq}p>1 {/eq} then the p-series converges.

If {eq}0<p\leq1 {/eq} then the p-series diverges.

Answer and Explanation:

We are given {eq}\displaystyle \sum_{n=0}^{\infty} \frac{1}{n^2 + 1} {/eq}

By comparison test:

Where {eq}\displaystyle \sum_{n=0}^{\infty}...

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from PSAT Prep: Tutoring Solution

Chapter 10 / Lesson 13

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