$$-\sum_{\color{red}{n=1}}^{\infty}nc_{n}x^{n}=-\sum_{\color{red}{n=0}}^{\infty}nc_{n}x^{n}$$
How come one starts at $1$ and the other starts at $0$ yet their equal? Do they both equal infinity?
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$\begingroup$ What is the value of the $0$-th term of the second sum? $\endgroup$– DarioNov 24, 2014 at 22:47
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$\begingroup$ Same reason $1+2+3+\cdots=0+1+2+\cdots$ $\endgroup$– JohnDNov 24, 2014 at 22:48
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1 Answer
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When $n=0$, $nc_nx^n$ is also zero. So the first sum just leaves that zero term out while the second one includes it. There is no change in the sum.
answered Nov 24, 2014 at 22:50