Power series practice problems

The following are practice problems for your own use. They will not be collected or marked; the idea is to provide more practice at the more computational aspects of the class, if desired, since the homeworks often focus on more theoretical aspects. I suggest focusing on problems targeted at aspects you feel less confident about.

Problem 1. Find the radius of convergence of ${\sum }_{n=0}^{\mathrm{\infty }}\frac{n^2}{3^n}{z}^n$.

Problem 2. Suppose the power series ${\sum }_{n=0}^{\mathrm{\infty }}{a}_n{z}^n$ has radius of convergence $R=4$. What is the radius of convergence of ${\sum }_{n=0}^{\mathrm{\infty }}{a}_n{z}^{2n}$?

Problem 3. Find the radius of convergence of the Taylor series of $f(z)=\frac{1}{z^2+2z+5}$ centered at $z_0=0$.

Problem 4. Find the Taylor series of $f(z)=\frac{1}{{(1-z)}^2}$ centered at $z_0=0$ by differentiating the geometric series term by term.

Problem 5. Let $f(z)=\frac{1}{z^2-3z+2}$. If $f(z)={\sum }_{n=0}^{\mathrm{\infty }}{a}_n{z}^{-n}$ is its series expansion at infinity, find the smallest real number $R$ such that the series converges on $\{z:|z|>R\}$.

Problem 6. Let $f(z)=z\sin (\pi z)$. What is the order of the zero at $z=0$? At $z=1$?

Problem 7. Determine the order of the zero of $f(z)=1-\cos (z^2)$ at $z=0$.

Problem 8. Let $f:D\to ℂ$ be an analytic function on a domain $D$ including the interval $[0,1]\subset ℝ$, and suppose $f(1/n)=0$ for all positive integers $n$. What can you conclude about $f$?

Problem 9. Using uniform convergence, justify that ${\int }_0^z\frac{1}{1+t}𝑑t={\sum }_{n=0}^{\mathrm{\infty }}\frac{{(-1)}^n{z}^{n+1}}{n+1}$ for .