Check the solution to "Discuss the convergence of the following series: ∑n=1∞cos⁡nπn2+1"

Josalynn

Answered question

2020-10-26

Discuss the convergence of the following series:

\sum_{n = 1}^{\infty} \frac{\cos n π}{n^{2} + 1}

Answer & Explanation

brawnyN

Skilled2020-10-27Added 91 answers

The given series is:
$\sum_{n = 1}^{\infty} \frac{\cos n π}{n^{2} + 1}$
The series can be written as: $\sum_{n = 1}^{\infty} \frac{\cos n π}{n^{2} + 1} = \sum_{n = 1}^{\infty} \frac{{(- 1)}^{n}}{n^{2} + 1}$
which is an alternating series with $a_{n} = \frac{1}{n^{2} + 1}$
Now,
$a_{n + 1} - a_{n}$
$= \frac{1}{{(n + 1)}^{2} + 1} - \frac{1}{n^{2} + 1}$
$= \frac{n^{2} + 1 - (n^{2} + 2 n + 2)}{(n^{2} + 1) ({(n + 1)}^{2} + 1)}$
$< 0$
$\Rightarrow a_{n + 1} < a_{n}$
$< 0$
So ${a_{n}}$ is decreasing.
And $lim_{n \to \infty} a_{n} = lim_{n \to \infty} \frac{1}{n^{2} + 1} = 0$
Therefore, from an alternating series test, the given series is convergent.

Jeffrey Jordon

Expert2021-12-27Added 2605 answers

Answer is given below (on video)

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