- #1
Chewy0087
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Homework Statement
Given that
[tex]\sum^{n}_{k=0} x^{k}= \frac{1-x^{n+1}}{1-x} [/tex]
Obtain a similar result for ;
[tex]\sum^{n}_{k=1} kx^{k}[/tex]
[tex]\sum^{n}_{k=1} k^{2}x^{k}[/tex]
The Attempt at a Solution
Hey, well basically my trouble with this question stems from the manipulation of the limits and the effects that it has on the series itself;
For the first one, I differentiated giving me;
[tex]\sum^{n}_{k=1} kx^{k-1}= \frac{x^{n+1}-x^{n}(n+1)+1}{(1-x)^{2}} [/tex]
Obviously when k<0 it doesn't hold as it would be 0, so I changed it to 1 from 0, and I can then reduce the lower index to 0 to make it into a better form.
However I'm just confused as to how to manipulate the index, and as to whether it would have any effect on the initial RHS equation. I don't seem to be getting the right answer in the way I've done above.
I have looked this up but no-where seems to be very helpful about it...
Thanks
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