(adsbygoogle = window.adsbygoogle || []).push({}); [solved] Sum of k x^k?

I happened upon a thread in a math forum, where someone asserted that this is true:

[tex]\sum_{k=0}^\infty (k+1) \left(\frac{5}{6}\right)^k = 36[/tex]

I suppose this makes intuitive sense. But if it's true, it must have a general form. I.e.,

[tex]\sum_{k=0}^\infty (k+1) r^k = ?[/tex]

Now, I know that the geometric series converges like so:

[tex]\sum_{k=0}^\infty r^k = \frac{1}{1-r}[/tex]

But by multiplying by (k+1) inside the summation completely changes things. Is there a name for this series? Is it true that it converges? If so, what does it converge to?

This question won't stop plaguing me. Since I don't know what this series is called, I'm having a hard time searching for it on the Internet.

Thanks!

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# Sum of k x^k?

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