Find a power series for $$f(z)=\int_{[0,1]}\frac{1}{1-wz}dw$$ What I did:
For $|wz|<1$, i.e. $|z|<\frac{1}{|w|}$, we have $$\frac{1}{1-wz}=\sum^{\infty}_{k=0}(wz)^k$$ So $$f(z)=\int^1_0 \sum^{\infty}_{k=0}(wz)^k dw=\sum^{\infty}_{k=0}\int^1_0(wz)^kdw=\sum^{\infty}_{k=0}z^k\frac{1}{k+1}w^{k+1}|^1_0=\sum^{\infty}_{k=0}\frac{z^k}{k+1}$$ Could anyone check my solutions? Since we need $|z|<\frac{1}{|w|}$ here, would there be a problem by using $w$ here?