Evaluate the series

**Jojo123** · November 15th 2008, 02:48 AM

1. sum of (n/2^n) from n=1 to infinite
2. sum of (n(n-1)x^n) from n=2 to infinite

**Soroban** · November 15th 2008, 04:09 AM

Hello, Jojo123!

Here's the first one . . .

$1)\;\;\sum^{\infty}_{n=1} \frac{n}{2^n}$

We are given: . $S \;=\;\frac{1}{2} + \frac{2}{2^2} + \frac{3}{2^3} + \frac{4}{2^4} + \frac{5}{2^5} +\hdots$

Multiply by $\tfrac{1}{2}\!:\;\;\frac{1}{2}S \;=\;\qquad \frac{1}{2^2} + \frac{2}{2^3} + \frac{3}{2^4} + \frac{4}{2^5} + \hdots$

$\text{Subtract: }\qquad\quad\frac{1}{2}S \;=\;\underbrace{\frac{1}{2} + \frac{1}{2^2} + \frac{1}{2^3} + \frac{1}{2^4} + \hdots}_{\text{geometric series}}$

The geometric series has a sum of $1.$

. . So we have: . $\frac{1}{2}S \:=\:1 \quad\Rightarrow\quad\boxed{ S \:=\:2}$

**Moo** · November 15th 2008, 08:30 AM

Originally Posted by Jojo123

2. sum of (n(n-1)x^n) from n=2 to infinite

$f(x)=\sum_{n=0}^\infty x^n=\frac{1}{1-x}$ (assuming |x|<1)

hence $f'(x)=\sum_{n=1}^\infty nx^{n-1}=\left(\frac{1}{1-x}\right)'=\frac{1}{(1-x)^2}$

$f''(x)=\sum_{n=2}^\infty n(n-1)x^{n-2}=\left(\frac{1}{(1-x)^2}\right)'=\frac{1}{2(1-x)^3}$

**Krizalid** · November 15th 2008, 09:38 AM

Originally Posted by Jojo123

2. sum of (n(n-1)x^n) from n=2 to infinite

Put $\alpha=\sum\limits_{n=1}^{\infty }{nx^{n}}\implies \alpha \cdot x=\sum\limits_{n=1}^{\infty }{nx^{n+1}}.$ From here it's not hard to see that $\alpha =\frac{x}{(1-x)^{2}}.$ Now put $\beta=\sum\limits_{n=1}^{\infty }{n(n-1)x^{n}}\implies \beta \cdot x=\sum\limits_{n=1}^{\infty }{n(n-1)x^{n+1}}$ and compute $\beta-\beta\cdot x.$

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