# Is there a formula to calculate the sum of this series?

Given that "r" is a constant real number, I have a series that goes like this:

$$r^1 + r^2 + r^3 ...$$

Is there a formula that can tell the sum of the first n elements? So basically a formula for calculating this sum:

$$\sum_{k=1}^{n} r^k$$

Thanks for any help!

• This is a geometric series, see mathworld.wolfram.com/GeometricSeries.html for how to compute the sum.
– Ben
Nov 11, 2015 at 23:12
• Ah, thanks. I had no idea it was called that. I'll check it out then :) Nov 11, 2015 at 23:13
• Duplicate of math.stackexchange.com/q/1185259
– user147263
Nov 11, 2015 at 23:14

This is the sum of a geometric series, with starting term $r$ and common ratio of $r$ as well.

Now, the general sum of the this series is:

$$S = \frac{1-r^{n+1}}{1-r}$$

A special case is when $|r| < 1$:

$$S = \frac{1}{1-r}$$

• What's the point of posting this for the $N$th time?
– user147263
Nov 11, 2015 at 23:15
• To assist users who can't perform a simple google search Nov 11, 2015 at 23:15
• I tried to google it, but since I had no idea that this was called a geometric series, I couldn't really find anything. Now that I know, I can find the answer myself of course. I just didn't know it was called that. But thanks anyway! Nov 11, 2015 at 23:18
• @NormalHuman you see the OPs answer. Nov 11, 2015 at 23:20
• My point stands: they can be assisted by marking their question as a duplicate of a canonical question, better than by a hurried two-line answer.
– user147263
Nov 11, 2015 at 23:20