What does the sum $$0(1/4)+1(3/4)(1/4)+2(3/4)^2(1/4)+3(3/4)^3(1/4)+⋯$$ equal?
I simplified it to $$3/16(1+2(3/4)+3(3/4)^2+ \cdots)$$ but now I'm stuck. Solutions are highly appreciated.
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1Hint: Geometric series – Jacky Chong Oct 18 '16 at 22:58
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1Compute $\sum_{k=0}^\infty {1 \over 4} x^k$ and differentiate. – copper.hat Oct 18 '16 at 22:59
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Let $$ S = \sum_{n\geq 0}\frac{n 3^n}{4^{n+1}}=\sum_{n\geq 1}\frac{n 3^n}{4^{n+1}} $$ and consider that $$ 4S = \sum_{n\geq 1}\frac{n 3^n}{4^n} = \sum_{n\geq 0}\frac{(n+1)3^{n+1}}{4^{n+1}} = 3\sum_{n\geq 0}\frac{(n+1) 3^n}{4^{n+1}}=3S+\sum_{n\geq 0}\frac{3^{n+1}}{4^{n+1}} $$ so that $$S = \sum_{n\geq 0}\frac{3^{n+1}}{4^{n+1}} = \frac{\frac{3}{4}}{1-\frac{3}{4}}=\color{red}{3}.$$
Jack D'Aurizio
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