Suppose $\int_{a}^{b} f(x,l) g(x) dx=0...(1)$ with Taylor series of $f(x)=\sum_{i=0}^{\infty} a_n x^n$, where $a_n$ depends upon parameter $l$. I want to construct non-zero bounded integrable function $g(x)$.
Since we are on finite interval, series $\sum_{i=0}^{\infty} a_n x^n g(x)$ converges to $f(x)g(x)$ uniformly(?), can I integrate term by term?
Then problem reduces to construction of $g(x)$ such that $\int_{a}^{b}x^n g(x) dx=0$ for $n=0,1,2...$ which I can do. My problem is with integrating term by term, can I construct $g(x)$ in this way?