Find a necessary and sufficient condition on $A$, $B$, and $C$ for which the series converges and find the sum in case of convergent. $$\sum_{n=0}^\infty \frac A{5n+1}+\frac B{5n+2}+\frac C{5n+3}$$ I found $A + B + C = 0$. Is that right?
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Can you show more of your works? – Oct 22 '13 at 03:46
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That is correct. To me the easiest way is to bring the expression to a common denominator (we do not have to write out all the terms of the numerator). – André Nicolas Oct 22 '13 at 03:52
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Nice question! (+1) Combining three divergent to make one convergent series in such a simple yet interesting fashion. – Paramanand Singh Oct 22 '13 at 04:05
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Note that $\forall n\ge1 \wedge \forall 1\leq j \leq 3$ $$\frac{1}{5n+j}=\frac{1}{5n}-\frac{j}{5n(5n+j)}=\frac{1}{5n}+O(\frac{1}{n^2})$$
So that we have, $\forall n\ge1$
$$\frac{A}{5n+1}+\frac{B}{5n+2}+\frac{C}{5n+3}=\frac{(A+B+C)}{5}\frac{1}{n}+O(\frac{1}{n^2})$$
So clearly we must have $A+B+C=0$ in order for your series to converge.
Ethan Splaver
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