Suppose $f(z)=\sum_{n=0}a_nz^n$ such that the radius of convergence is 1 and $a_0\neq0$.Let $s_n(z)=a_0+a_1z+\cdots+a_nz^n$ be the partial sums.Let $1>\delta>0$ such that $f\neq 0$ on $|z|=1-\delta$.Prove that there is $K>0$ and integer N such that for all $n\ge N$ the modulus of the product of zeros of $s_n(z)$ inside $|z|=1-\delta$ is at least $K$.
We know that $f(0)\neq0$,what bothers me is that since for sufficiently large $n$,the zeros of $s_n(z)$ should be near $f$,what if $f$ has no zeros inside $|z|=1-\delta$.The reason to choose a smaller circle than the unit one seems to avoid the zeros of $s_n$ goes to the boundary.I sense that I shall invoke the radius of convergence to derive some constraints on the coefficients so as to ensure polynomial $s_n$ will have zeros inside the circle,which is near zero.