Let $p(x)$ be a polynomial of the real variable $x$ of degree $k\geq 1$ .Consider the power series $$f(z)=\sum_{n=0}^{\infty}p(n)z^n$$ where z is a complex variable .Then the radius of convergence of $f(z)$ is ?

  • $\begingroup$ Use Ratio Test or Root Test. $\endgroup$ – André Nicolas Apr 21 '16 at 2:20
  • $\begingroup$ $(\limsup_{n\rightarrow \infty} |p(n)|^{1/n})^{-1}$ $\endgroup$ – Rick Sanchez Apr 21 '16 at 2:29
  • $\begingroup$ what is the radius of convergence of $\sum_n n^k z^n$ ??? $\endgroup$ – reuns Apr 21 '16 at 2:40

$\lim_{n\rightarrow \infty} \frac{p(n+1)}{p(n)}=1$ for any polynomial of degree $k \ge 1$.Therefore radius of convergence is 1 for given power series.


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