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

Question

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 '15 at 23:12
Ah, thanks. I had no idea it was called that. I'll check it out then :) — adam10603, Nov 11 '15 at 23:13

Varun Iyer · Accepted Answer · 2015-11-11 23:14:40Z

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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 $|r| < 1$:

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

answered Nov 11 '15 at 23:14

Varun Iyer

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$\begingroup$ What's the point of posting this for the $N$th time? $\endgroup$ – user147263 Nov 11 '15 at 23:15
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$\begingroup$ To assist users who can't perform a simple google search $\endgroup$ – Varun Iyer Nov 11 '15 at 23:15
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$\begingroup$ 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! $\endgroup$ – adam10603 Nov 11 '15 at 23:18
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$\begingroup$ @NormalHuman you see the OPs answer. $\endgroup$ – Varun Iyer Nov 11 '15 at 23:20
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$\begingroup$ 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. $\endgroup$ – user147263 Nov 11 '15 at 23:20

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