How would I go about deriving the value of the following infinite sum: $\sum\limits_{k=1}^\infty kx^k$ ?
I thought about expanding first: $\sum\limits_{k=1}^\infty kx^k= x + 2x^2 + 3x^3 + \cdots$
Then a bit of algebra: $\sum\limits_{k=1}^\infty kx^k - \sum\limits_{k=1}^\infty (k-1)x^k = x + x^2 + x^3 + \cdots + 1 -1 $
And now I'm stuck with this: $\sum\limits_{k=1}^\infty x^k = \frac{x}{1-x}$
How can I introduce the $k$ into $\sum\limits_{k=1}^\infty x^k$ ? Or is there a different approach that I don't know of?
Any help is much appreciated.
Answer
As you know $\sum_{k=0}^\infty x^k$ you can differentiate the result. Justify that you can differentiate the series term-by-term.
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