# Infinite(simal) Sum

Have fun!

Find a formula for $$\sum_{i=0}^\infty i^{2014}x^{i}$$. In other words, find $$0^{2014}x^{0}+1^{2014}x^{1}+2^{2014}x^{2}+3^{2014}x^{3}+4^{2014}x^{4}+...$$.

Hint: Try a form of recursion(not with numbers, but with a formula).

Note by Tristan Shin
4 years, 5 months ago

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We can get a recurrence relation in the following manner:

For positive integers k we define $$T_{k}(x)=\sum_{i=0}^{\infty }i^kx^i$$ $$\forall x \in (0,1)$$

We know $$T_{1}(x)=\frac{x}{(1-x)^2}$$

Also $$T_{k+1}(x)=x\frac{dT_{k}(x)}{dx}$$

- 4 years, 5 months ago

To be precise, is there a way of expanding your recursion formula?

- 4 years, 5 months ago

Sorry, I didnt get you. What do you exactly mean by expanding the recursion formula? Did you mean this-

$$T_{k+1}(x)=T_{k}(x)+x^2T_{k-1}''(x)$$

- 4 years, 5 months ago

I found a simpler recursion formula that only reaches back one term.

For a start, I labeled $$T_{k}=\frac {f(x)}{(1-x)^{k+1}}$$. Then, try working around with that.

- 4 years, 5 months ago

Just a note, I know how to do it, but it would take a very long time(probably over a month straight) of working out some functions. I'm not really looking for an exact solution(don't actually take the time to do it), but if anyone can discover the method I'm using, that would be wonderful.

- 4 years, 5 months ago