To "solve" this problem, I will consider the following geometric series, of course, |x|<1:
This answers the question for , to solve this for other , I will do the derivative of both sides, before that, I will use the following "formula", to help get things done faster(I used the Power Rule and the Chain Rule to reach this result without too much work):
Now I will multiply everything by , to adjust the right side of the equality:
I will apply partial fractions on the left side of the equality, using the following formula(you can prove it by simply doing the operations):
Then I get:
Note to get the answer for , I can do the derivative of both sides, then multiply by , and then apply partial fractions, but then I can do for and so on;
Does anyone know how to solve the recurrence relation I get?
Let's suppose that I know the answer for , there is a sequence of as the coefficients of such that:
I will do the derivative of both sides:
Then I will multiply both sides by :
Applying partial fractions:
This is the solution for , therefore I can say that this is equal to the sum of , with the appropriate coefficients , also, I will take out the last and first coefficients because they don't have pairs, with that I can say:
With this, I can say some things:
There are some patterns I can see(they can be proved by induction):
But I can't give a complete closed "simple" formula, like Faulhaber's Formula, does anyone know or has a suggestion to tackle this problem?