power series!

• May 28th 2013, 05:51 AM
lawochekel
power series!
$\sum_{n=1}^{\infty} x^n$

pls i don't have any ideal of finding the sum-function of the above series, help.

thanks
• May 28th 2013, 05:57 AM
Plato
Re: power series!
Quote:

Originally Posted by lawochekel
$\sum_{n=1}^{\infty} x^n$

find the sum-function of the above series

Provided that $|x|<1~\&~J\in\mathbb{Z}$ then $\sum_{n=J}^{\infty}{a x^n}=\frac{ax^J}{1-x} ~.$
• May 28th 2013, 06:24 PM
Prove It
Re: power series!
Under the conditions given by Plato, this would be an infinite geometric series. You should read more about geometric sequences and series.
• May 29th 2013, 01:54 AM
Kiwi_Dave
Re: power series!
$\sum_{n=1}^{\infty} x^n=x + \sum_{n=2}^{\infty} x^n=x + x\sum_{n=1}^{\infty} x^n$

we have the original sum again on the right hand side (multiplied by x). So

$\sum_{n=1}^{\infty} x^n=x + x\sum_{n=1}^{\infty} x^n$

So

$(1-x)\sum_{n=1}^{\infty} x^n=x$

finally as others have said:

$\sum_{n=1}^{\infty} x^n=\frac{x}{1-x}$