# How do you determine if the series the converges conditionally, absolutely or diverges given Sigma (cos(npi))/(n+1) from [1,oo)?

Apr 1, 2017

${\sum}_{n = 1}^{\infty} \frac{\cos \left(n \pi\right)}{n + 1}$ converges conditionally.

#### Explanation:

Since $\cos \left(n \pi\right) = {\left(- 1\right)}^{n}$,

${\sum}_{n = 1}^{\infty} \frac{\cos \left(n \pi\right)}{n + 1} = {\sum}_{n = 1}^{\infty} \frac{{\left(- 1\right)}^{n}}{n + 1}$

Let us see if it is absolutely convergent.

${\sum}_{n = 1}^{\infty} \left\mid \frac{{\left(- 1\right)}^{n}}{n + 1} \right\mid = {\sum}_{n = 1}^{\infty} \frac{1}{n + 1} = \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \cdots$,

which is a harmonic series (divergent). So, it is NOT absolutely convergent.

Let us see if it is conditionally convergent.

Since $\frac{1}{n + 1}$ is decreasing and ${\lim}_{n \to \infty} \frac{1}{n + 1} = 0$, by Alternating Series Test, we know that the series is convergent.

Hence, the series is conditionally convergent.