How to prove that the sum of a convergent geometric series of the form $1 + r + r^2 .... + r^n > 1/2$?

Question

I am trying to prove that the sum of a convergent geometric series of the form \begin{equation*} 1 + r + r^2 .... + r^n > \frac{1}{2} \end{equation*} but I have no idea how to go about this. Looking at the series, it is pretty obvious, but I just don't know how to prove that this is the case.

If someone wouldn't mind explaining how I would go about proving this, I would be very grateful.

Thanks.

Corey :)

Answer 1

HINT: your inequality is equivalent to $\frac{r^{n+1}-1}{r-1}>\frac{1}{2}$

Answer 2

Whether it converges or not depends on the value of r compared to 1.Follow the hint given by Dr.Sonnard and examine the different cases for $∣r∣<1$ and $∣r∣≥1$.

When you are done with that,you can try looking at the general case of $a+ar+ar^2+...+ar^n+...$ The approach to the answer is identical.