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

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Answer 1 · 2017-04-01T06:58:06

$sum_{n=1}^infty{cos(n pi)}/{n+1}$ converges conditionally.

Since $cos(npi)=(-1)^n$ ,

$sum_{n=1}^infty{cos(n pi)}/{n+1}=sum_{n=1}^infty{(-1)^n}/{n+1}$

Let us see if it is absolutely convergent.

$sum_{n=1}^inftyabs{{(-1)^n}/{n+1}}=sum_{n=1}^infty1/{n+1}=1/2+1/3+1/4+cdots$ ,

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

Let us see if it is conditionally convergent.

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

Hence, the series is conditionally convergent.

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