### Author Topic: Q5 TUT 5301  (Read 2910 times)

#### Victor Ivrii

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##### Q5 TUT 5301
« on: November 02, 2018, 03:34:33 PM »
Find the power-series expansion about the given point for the given function; find the largest disc in which the series is valid:
$$\frac{z+2}{z+3}\qquad\text{about}\; z_0 = -1.$$

#### Tianfangtong Zhang

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##### Re: Q5 TUT 5301
« Reply #1 on: November 02, 2018, 03:43:00 PM »
\begin{align*}
\frac{z+2}{z+3} &= \frac{z+3-1}{z+3} \\ &= 1-\frac{1}{z+3}\\
&= 1- \frac{1}{z+1+2} \\ &= 1 - \frac{1}{2} \frac{1}{1+\frac{z+1}{2}}\\
&= 1 - \frac{1}{2} \frac{1}{1-\frac{-(z+1)}{2}}\\
&= 1 - \sum_{n=0}^{\infty}(\frac{-(z+1)}{2})^n \\
&= 1 - \sum_{n=0}^{\infty}\frac{(-1)^n}{2^{n+1}}(z-(-1))^n
\end{align*}

#### Ende Jin

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##### Re: Q5 TUT 5301
« Reply #2 on: November 03, 2018, 02:29:15 PM »
For the former one, the largest disc is $\{z: |z + 1| < 2\}$

For the latter one, the largest disc is the whole complex plane.
« Last Edit: November 04, 2018, 09:41:37 PM by Victor Ivrii »