Power series problems

Question

How can I find the power series expansion of following functions. I don't have any idea, these seem intimidating. Please help how to proceed.

1. $(x+\sqrt{1+x^2})^a$

2. $\sqrt{\frac{1-\sqrt{1-x}}{x}}$

Answer 1

The derivative of the first expression, call it $u(x)$, is $$ u'(x)=\frac{a(x+\sqrt{1+x^2})^a}{\sqrt{1+x^2}} $$ so that $$ u'(x)=au(x)(1+x^2)^{-\frac12} $$ The binomial power series $(1+x^2)^{-1/2}=\sum c_mx^{2m}$ is known so that this last equation becomes, after comparing coefficients of equal degree, a triangular linear system for the power series coefficients of $u(x)$, $$ u_k=\frac{a}{k}\sum_{0\le m<k/2} u_{k-1-2m}c_m $$

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For the second equation note that by binomial identities $$ \sqrt{\frac{1-\sqrt{1-x}}{x}}=(1+\sqrt{1+x})^{-1/2} $$ which possibly will simplify the expansion, it should be possible to apply similar steps as above.