# Tagged Questions

43 views

### Prove that $\frac{1}{\sqrt{1-z}}=\sum_{n=0}^{\infty}\frac{1}{4^{n}}\binom{2n}{n}z^{n}$ using Cauchy product

need to prove using Cauchy product for series for all $\left|z\right|<1$ that $$\frac{1}{\sqrt{1-z}}=\sum_{n=0}^{\infty}\frac{1}{4^{n}}\binom{2n}{n}z^{n}$$ (with appropriate branch of the root ...
14 views

### Taylor expansion and expansion in powers of z-1

I am trying to expand $z^2/(z+1)^2$ as a Taylor Series. I have acquired its partial fraction decomposition of $z^2/(z+1)^2$ = $(1/6)*(1/(z+1)) + (5/6)(1/(z-5))$. The first term is in the form ...
40 views

36 views

### Radius of Convergence of product of power series

Is the following statement true? If $P(z)$ is a power series over $\mathbb C$, then $P(z)$and $P(z)^n$ have same radius of convergence for any positive integer n.
59 views

### Complex Power series with factorials

Find the radius of convergence of $$\displaystyle\sum_{n=0}^\infty z^{n!}$$ $$\displaystyle\sum_{n=1}^\infty {(-1)^nz^{n(n+1)}}/{n}$$ What is the behavior of the series for $z=1, -1,i$
32 views

### Convergence of complex power series question

I need some help to solve this problem and find the domain of convergence of the following power series: $$\displaystyle\sum_{n=0}^\infty(2^n+i^n)(z-2i)^n$$ Thank you!
34 views

### Radius of convergence of the given power series in powers $(z+2)^n-1$.

$$\sum_{n=0}^{\infty}\frac{(z+2)^{n-1}}{(n+1)^24^n}$$ Please help me solve this question on finding the radius of convergence..
26 views

### Close form of a power series starting at $n=2$

This is the power series I am looking at $\sum_{n=2}^{\infty}{n(n-1)z^n}$. I want to find the closed form of this power series. This is my approach, if I divide the power series by $z^2$, then I ...
182 views

### Find a closed form from the given power series

I have the power series $\sum_{n=0}^{\infty} {z^{2n}\over{n!}}$, how do I find the closed form for this power series. I am aware that $e^z=\sum_{n=0}^{\infty} {z^{n}\over{n!}}$, so I tried to ...