Power series expansion of $x\ln(\sqrt{4+x^2}-x)$

Question

Find $a_n $ where $x \ln(\sqrt{4+x^2}-x) =\sum_{n=0}^{\infty} a_nx^n$.

I know that I must find power series expansion of $\ln(\sqrt{4+x^2})$ but it doesn't help. Can anyone give me a hint? many thanks

Answer 1

The factor $x$ can be momentarily disregarded; consider $f(x)=\ln(\sqrt{4+x^2}-x)$ and note that $$ f'(x)=\frac{\dfrac{x}{\sqrt{4+x^2}}-1}{\sqrt{4+x^2}-x}=-\frac{1}{\sqrt{4+x^2}}=-\frac{1}{2}(1+(x/2)^2)^{-1/2} $$ The Taylor development of the derivative can be written down. Integrate and multiply by $x$. $$ f'(x)=-\frac{1}{2}\sum_{n\ge0}\binom{-1/2}{n}\frac{x^{2n}}{2^{2n}} $$ So $$ f(x)=\ln2-\frac{1}{2}\sum_{n\ge0}\binom{-1/2}{n}\frac{x^{2n+1}}{2^{2n}(2n+1)} $$ I leave to you determining $a_n$.