<s>[INST] Calculate $\sum_{n=1}^{\infty}{c_n\over n}$ when $c_1=2, c_2=1, c_3=-3, c_{n+3}=c_n, n\in \mathbb{N}$ Let $(c_n)$ be defined as $c_1=2, c_2=1, c_3=-3, c_{n+3}=c_n, n\in \mathbb{N}$. I want to get $\sum_{n=1}^{\infty}{c_n\over n}$. I would appreciate any suggestions how to start on this. It is my suspicion that I would have to resort to integration.
 [/INST] Break the sum into three pieces
\begin{eqnarray*}
S=\sum_{n=0} ^{\infty} \left( \frac{2}{3n+1} + \frac{1}{3n+2} +\frac{-3}{3n+3} \right) \\
\int_0^1 x^n dx = \frac{1}{n+1} 
\end{eqnarray*}
Now use the above formula ... interchange the sum & integral & one has ... 
\begin{eqnarray*}
S= \int_0^{1} \frac{2+x-3x^2}{1-x^3} dx =\int_0^{1} \frac{2+3x}{1+x+x^2} dx =\frac{\pi}{6 \sqrt{3}} + \frac{3}{2} \ln 3. 
\end{eqnarray*}
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