- 369

- 0

**1. The problem statement, all variables and given/known data**

Given that

[tex]\sum^{n}_{k=0} x^{k}= \frac{1-x^{n+1}}{1-x} [/tex]

Obtain a similiar result for ;

[tex]\sum^{n}_{k=1} kx^{k}[/tex]

[tex]\sum^{n}_{k=1} k^{2}x^{k}[/tex]

**3. 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

Last edited: