two series questions

**kuntah** · February 26th 2008, 10:15 AM

Hi all,

I had two questions

1. $\sum ^{\infty}_{i=1} \frac{1}{\left(i-1\right)!} \frac{d}{d\lambda} \left(\lambda^{i}\right)= \frac{d}{d\lambda} \sum ^{\infty}_{i=1} \frac{1}{\left(i-1\right)!} \lambda^{i}$

2. $\sum^{\infty}_{x=1} x r^{x} = \frac{r}{\left(1-r\right)^{2}}$

why are 1 and 2 true
For the first I think it has to do with the radius of convergence.

Can someone help to prove this,

Thank you very much

**Peritus** · February 26th 2008, 10:28 AM

1. Power series - Wikipedia, the free encyclopedia

2.

$\sum\limits_{x = 1}^\infty {xr^x } = r\sum\limits_{x = 1}^\infty {xr^{x - 1} = } r\sum\limits_{x = 1}^\infty {\frac{d} {{dr}}r^x = } r\frac{d} {{dr}}\sum\limits_{x = 1}^\infty {r^x = } r\frac{d} {{dr}}\frac{r} {{1 - r}} = \frac{r} {{\left( {1 - r} \right)^2 }} $

**kuntah** · February 26th 2008, 10:36 AM

thanks for the help

I already solved the first now and thanks for the seccenodn )it was the same mechanism

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