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The series is $\displaystyle \sum_{n=1}^{+\infty}a_n=\displaystyle\sum_{n=1}^{+\infty}c$. This means that the mth partial sum is $S_m=a_1+a_2+\ldots+a_m=mc$ so, $$S=\lim_{m\to \infty}S_m=\lim_{m\to +\infty}mc=\left \{ \begin{matrix}{ +\infty}&\mbox{ if }& c>0\\-\infty & \mbox{if}& c<0\\0 & \mbox{if}& c=0\end{matrix} \right.$$ As a consequence the series is convergent if and only if $c=0$.