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Caspian
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[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!
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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