find the sequence of polynomials $(P_n)$ such that $\sum P_n$ converges absolutely (that is $\sum \|P_n\|_{\infty}\lt\infty $) but is not convergent in the space ($\mathcal{P}[0,1], \|.\|_{\infty}$, i.e. sup norm)
i thought of the taylor series of cosx, $e^x$ etc. the $(P_n)$ is not convergent in the space of polynomials but are they absolutely convergent to a polynomial?
could you please give some insight
You can do it with the Taylor series of $\log(1+x)$, in which case $$ P_n(x)=\sum_{k=1}^n(-1)^n\,\frac{x^n}{n}. $$