Toronto Math Forum
MAT3342018F => MAT334Tests => Quiz5 => Topic started by: Victor Ivrii on November 02, 2018, 03:34:33 PM

Find the powerseries expansion about the given point for the given function; find the largest disc in which the series is valid:
$$\frac{z+2}{z+3}\qquad\text{about}\; z_0 = 1.$$

\begin{align*}
\frac{z+2}{z+3} &= \frac{z+31}{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*}

For the former one, the largest disc is $\{z: z + 1 < 2\}$
For the latter one, the largest disc is the whole complex plane.