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I am approximating certain solution of an ODE by power expansions. As it is customary, I propose an ansatz and then I check for the coefficients to satisfy the ODE. At some point of my computations I get something as follows (I am expanding with respect to the variable $u$)

$$ \sum_{k=0}^\infty a_{k-2}(u+1)^{7/2}u^{k/2}=\sum_{k=0}^\infty a_ku^{k/2}. $$

So as I have understood I have to collect the coefficients of the same power of $u$. But I do not know how to handle the term $(u+1)^{7/2}$. In principle I have

$$ a_{k-2}(u+1)^{7/2}=a_k $$

but the coefficients shouldn't depend on $u$ right?

Any help is appreciated.

share|improve this question
Ok. I think I've got it. in the series, first I write $(u+1)^{7/2}$ in its binomial expansion (which I did't know holds for arbitrary complex exponents). After that I am able to play a little bit with the exponents, re-arrange, shift, and finally collect common coefficients. –  PepeToro Nov 6 '13 at 14:58

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