0
votes
0answers
28 views

Taylor expansion with complex valued number

I like to do a Taylor expansion with a complex valued number, e.g.: $f(z) = \frac{1}{(1-\mathrm{i}z)} \qquad z \in \mathbb{C}$ Is there any restriction to expand this (let's say around $z=0$) ...
2
votes
1answer
212 views

Intuition regarding Taylor series for $\frac{e^z}{1-3z}$.

The question asks me to find the Taylor series for $$f(z)=\frac{e^z}{1-3z}.$$ The radius of convergence is $|z|<1/3$ and I know the expansions for $e^z$ and $1/(1-3z)$ are \begin{align} e^z ...
0
votes
1answer
391 views

Taylor Series Expansion with e and sin

Show that when $z\neq0$, (a) $$\frac{e^z}{z^2}=\frac{1}{z^2}+\frac{1}{z}+\frac{1}{2!}+\frac{z}{3!}+\frac{z^2}{4!}+...$$ (b) ...
0
votes
1answer
629 views

Deriving Taylor series for function from geometric series

Given the geometric series $\frac{1}{1-z} = \sum_{n=0}^n = 1 + z + z^2 + ...$ If there is a function $f(z)=\frac{1}{z+j}$ how would you get it's Taylor series about center z = 1? I have tried the ...
2
votes
2answers
124 views

Need a hint with this question

I'm looking over one of my past papers and I'm having some trouble with the following question. By considering the series expansion of: $\ln(1-z)$, where $z=\frac{e^{i\theta}}{2}$, show that ...
11
votes
2answers
413 views

A property of roots of the truncated series for $\sin(x)$

Let $p_n(x) = \sum\limits_{k=0}^n \frac{(-1)^kx^{2k+1}}{(2k+1)!}$ In other words, $p_n$ is the polynomial made of the first $n$ terms of the Taylor expansion of $\sin(x)$ around $x = 0$. ...