Deriving formula for sums of any power

[Darya]

I'm going through a book The Art of proof and I've stumbled upon this problem.
Now, I know how to derive a formula for sum [MATH]j^m, m=1,2,3,4,5,..n[/MATH] for any power m using telescoping sums but not this way.
What I've done do far is just differentiating
I got [MATH] Σ jx[/MATH]^(j-1) from j=1 to k on the left side.
And [MATH]((1-x^(k+1))-x^k(k+1)(1-x))/(1-x)^2[/MATH].
I have no idea how to then use L' Hopitals rule as well as how to use Math module on this forum.
Any hints?

 

[Cubist]

I have no idea how to then use L' Hopitals rule

I'm not 100% sure that this is the intended method, but if you now work out the limit as x tends to 1 then you should end up with a formula for \( \sum_{j=0}^kj\), ie the formula for when m=1. And this method could be extended by differentiating both sides twice before taking the limit x->1, and then you should end up with a formula for m=2

 

[Darya]

I'm not 100% sure that this is the intended method, but if you now work out the limit as x tends to 1 then you should end up with a formula for \( \sum_{j=0}^kj\), ie the formula for when m=1. And this method could be extended by differentiating both sides twice before taking the limit x->1, and then you should end up with a formula for m=2

THANK YOU SO MUCH!!