Thursday, 29 March 2018

sequences and series - Find value of infinite sum











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=1kxkk=1(k1)xk=x+x2+x3++11



And now I'm stuck with this: k=1xk=x1x




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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