MHB Hair Canada's question at Yahoo Answers (Constant series)

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The series defined by a_n = c diverges unless c equals zero. For c greater than zero, the series diverges to positive infinity, while for c less than zero, it diverges to negative infinity. The mth partial sum is represented as S_m = mc, which leads to the conclusion that the series converges only when c equals zero. This indicates that the behavior of the series is directly tied to the value of c. Understanding this relationship is crucial for determining the convergence or divergence of the series.
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Here is the question:

no matter what value I put for n, i keep getting c. So my best guess is that this is a diverging sequence?

Here is a link to the question:

Math: Is a_n = c diverging or converging? - Yahoo! Answers

I have posted a link there to this topic so the OP can find my response.
 
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Hello Hair Canada,

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 $c=0$.
 
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