## What kind of series is $\sum_{n=0}^{\infty}{n\over{2^n}}$ ? [closed]

$\sum_{n=0}^{\infty}{n\over{2^n}}$

This series appears to converge to 2, but what kind of series is it?

(Source: "using mathematical formulas" in http://www.mytechinterviews.com/boys-and-girls)

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I don't know what exactly you mean by "appears", so you might already know this, but Maple 11 says the series does indeed converge to 2. – Ricky Demer Sep 16 2010 at 7:10
Convergent. I've seen such series called "arithmetic-geometric," but I don't think that nomenclature is commonplace. – Gerry Myerson Sep 16 2010 at 7:13
This is hardly an MO-level question. If you want a name for it, it's an infinite version of what was called an arithmetic-geometric progression in my O-level days. You can sum it as $f(1/2)$ where $f(x)=\sum nx^n$. Now $f(x)$ is more-or-less the derivative of an even easier series. – Robin Chapman Sep 16 2010 at 7:13
or the square, also – Pietro Majer Sep 16 2010 at 8:35