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# Infinite(simal) Sum

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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
2 years, 9 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 $$T_{k+1}(x)=x\frac{dT_{k}(x)}{dx}$$ · 2 years, 9 months ago

To be precise, is there a way of expanding your recursion formula? · 2 years, 9 months ago

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

$$T_{k+1}(x)=T_{k}(x)+x^2T_{k-1}''(x)$$ · 2 years, 9 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. · 2 years, 9 months ago