How would I go about deriving the value of the following infinite sum: ∞∑k=1kxk ?
I thought about expanding first: ∞∑k=1kxk=x+2x2+3x3+⋯
Then a bit of algebra: ∞∑k=1kxk−∞∑k=1(k−1)xk=x+x2+x3+⋯+1−1
And now I'm stuck with this: ∞∑k=1xk=x1−x
How can I introduce the k into ∞∑k=1xk ? Or is there a different approach that I don't know of?
Any help is much appreciated.
Answer
As you know ∑∞k=0xk you can differentiate the result. Justify that you can differentiate the series term-by-term.
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