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.