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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).

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## Comments

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

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

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

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

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

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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.

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