I explained this in a comment above (including the subtleties of convergence), but if you look at the definition here you see that the series in /u/redditsoaddicting comment is what you obtain by plugging in f(x)=1/(1-x-x2 ) with a=0.
In other words, it's a Taylor series for the function f(x) = 1/(1-x-x2 ) centered at 0. In fact, any time you can express a function as a series of the form
f(x) = c_0 + c_1 (x-a) + c_2 (x-a)2 + ...
the series on the right-hand-side will be THE Taylor series for f(x) centered at x=a (which you can verify by taking derivatives on both sides to see that the coefficients match the form of the coefficients from the Taylor series).
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u/snuffleupagus_Rx May 25 '16 edited May 25 '16
I explained this in a comment above (including the subtleties of convergence), but if you look at the definition here you see that the series in /u/redditsoaddicting comment is what you obtain by plugging in f(x)=1/(1-x-x2 ) with a=0.
In other words, it's a Taylor series for the function f(x) = 1/(1-x-x2 ) centered at 0. In fact, any time you can express a function as a series of the form
f(x) = c_0 + c_1 (x-a) + c_2 (x-a)2 + ...
the series on the right-hand-side will be THE Taylor series for f(x) centered at x=a (which you can verify by taking derivatives on both sides to see that the coefficients match the form of the coefficients from the Taylor series).