$\sum_{n=1}^{m}nr^{n}-r\sum_{n=1}^{m}nr^{n}=-mr^{m+1}+\sum_{n=1}^{m}r^{n}$ ,where $n,m$ are integers.
Is it true?
If yes how to show it?
Thank you.
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\begin{align*} \sum_{n=1}^{m}nr^{n}-r\sum_{n=1}^{m}nr^{n} &= \sum_{n=1}^{m}nr^{n}-\sum_{n=1}^{m}nr^{n+1} \\ &= r + \sum_{n=2}^{m}nr^{n}-\sum_{n=2}^{m}(n-1)r^{n} - m r^{m+1} \\ &= r + \sum_{n=2}^m (n - (n-1))r^n - mr^{m+1}\\ &= r + \sum_{n=2}^m r^n- mr^{m+1}\\ &= \sum_{n=1}^m r^n- mr^{m+1} \end{align*}
S L
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Thanks. I think in the second step it should be $-mr^{m+1}$ instead of $-mr^{m+2}$. – Silent Sep 24 '13 at 08:15
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@Sush thanks for correction ... i'll edit it. – S L Sep 24 '13 at 08:15