Assistance with series, please

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The discussion focuses on computing the sum from k=0 to infinity of the series (k+1)(x)(1-x)^k. The user identifies a resemblance to geometric series and references the power series for 1/x, specifically (1-x)^k. A key insight provided is the differentiation technique, where (k+1)(1-x)^k can be expressed as -\frac{d}{dx}(1-x)^k, which is crucial for simplifying the series. This approach leads to a clearer path for solving the problem.

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Compute the sum from k=0 to infinity of (k+1)(x)(1-x)^k.

I don't have any ideas about how to start this one, except that perhaps it resembles a geometric series? Also we're supposed to use the power series for 1/x which we know to be (1-x)^k. (That's all I could figure out on my own so far) :confused: What I could really use some help with is how to get started, then I should be able to pick it up from there. Thank you SO much for any help!
 
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hint: [itex](k+1)(1-x)^k = -\frac{d}{dx}(1-x)^k[/itex]
 

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