1. ## Geometric Sum

Hello all,

Just a quick question, if you are summing r^t (where r is between zero and one, and you are summing across t to infinity), this converges to 1/(1-r). BUT if you don't start at 0, but you start at some value of t (lets call it t*), is there any nice way to express what this converges to?

Thanks for the help,

Nick

2. Originally Posted by salohcin
Hello all,

Just a quick question, if you are summing r^t (where r is between zero and one, and you are summing across t to infinity), this converges to 1/(1-r). BUT if you don't start at 0, but you start at some value of t (lets call it t*), is there any nice way to express what this converges to?

Thanks for the help,

Nick
$|r| < 1$

$\sum_{t = 0}^{\infty} r^t = 1 + r + r^2 + ... = \frac{1}{1-r}$

$\sum_{t = 1}^{\infty} r^t = r + r^2 + ... = \frac{1}{1-r} - 1 = \frac{r}{1-r}$

3. Originally Posted by salohcin
if you are summing r^t (where r is between zero and one, and you are summing across t to infinity), this converges to 1/(1-r). BUT if you don't start at 0, but you start at some value of t (lets call it t*), is there any nice way to express what this converges to?
If $N\in\mathbb{Z}~\&~|r|< 1$ then $\sum\limits_{k = N}^\infty {r^k } = \frac{{~r^N }}{{1 - r}}$