# geometric series problem

• Jul 7th 2011, 06:48 AM
mastermin346
geometric series problem
Given the infinite series$\displaystyle \sum_{r=0} ^\infty\left ( \frac{1-x}{x} \right )^r$

a)show that the series is a geometric series

b)determine the set of value of $\displaystyle x(x>0)$ so that the sum to infinity ,$\displaystyle S_{\infty}$ exist

c)if $\displaystyle x = 2$, find the sum of the first five terms,$\displaystyle S_5$ and $\displaystyle S_{\infty}$.Show that $\displaystyle S_5 : S_{\infty} = 33:32$
• Jul 7th 2011, 07:15 AM
emakarov
Re: geometric series problem
This problem is solved in a standard way, so you need to show some effort.

$\displaystyle \sum_{r=0}^\infty a^r$ converges iff $\displaystyle |a|<1$. When you are looking for the set of x for which the series converges, remember that an equality changes sign when multiplied by a negative number.
• Jul 7th 2011, 07:20 AM
mastermin346
Re: geometric series problem
what should i do sir?
• Jul 7th 2011, 07:32 AM
Plato
Re: geometric series problem
Quote:

Originally Posted by mastermin346
what should i do sir?

Show some effort.
What is the formula for $\displaystyle S_5 = \sum\limits_{k = 0}^5 {r^k }~?$
• Jul 7th 2011, 07:32 AM
emakarov
Re: geometric series problem
For part (a), study the definition of geometric progression and geometric series.

For part (b), I gave you a hint. I used the term "converges." By definition, a series converges iff its sum to infinity exists.