Prove a + ar + ar^2 + + ar^n < a/r - 1

[Fairy111]

Homework Statement

Show that, if a>0 and 0<r<1, then for all n>=1 , a+ar+ar^2+...+ar^n < a/r -1

Homework Equations

The Attempt at a Solution

could someone please give me some hints as to how to start this questions. thanks

 

[Sisplat]

Hello,

http://en.wikipedia.org/wiki/Geometric_progression

The series you have is a geometric progression, follow the derivation for the sum of a geometric progression on the wikipedia page. Once you have derived the sum take the limit as n goes to infinity.

If this is a/(1 - r), or less, you have shown that the sum can never be greater than a/(1-r).

To take the limit you will also need to explicitly show that the series converges for some range of r.