$\displaystyle \sum_{n=1}^{\infty} x^n $
pls i don't have any ideal of finding the sum-function of the above series, help.
thanks
$\displaystyle \sum_{n=1}^{\infty} x^n=x + \sum_{n=2}^{\infty} x^n=x + x\sum_{n=1}^{\infty} x^n$
we have the original sum again on the right hand side (multiplied by x). So
$\displaystyle \sum_{n=1}^{\infty} x^n=x + x\sum_{n=1}^{\infty} x^n$
So
$\displaystyle (1-x)\sum_{n=1}^{\infty} x^n=x $
finally as others have said:
$\displaystyle \sum_{n=1}^{\infty} x^n=\frac{x}{1-x} $