Q4P - Mathematical Methods in Physical Sciences

Question

Find the disk of convergence for each of the following complex power series.

 $\underset{\mathbf{ }}{\mathbf{\sum }{\mathbf{z}}^{\mathbf{n}}}$

Step-by-Step Solution

Answer

The region of convergence is $\left|\mathrm{z}\right|<1$.

Step 1: Given Information.

The given power series, i.e., ${\mathrm{S}}_{\mathrm{n}}=\underset{ }{\sum {\mathrm{z}}^{\mathrm{n}}}$  

Step 2: Definition of Region of convergence.

Disc of convergence is defined as the interior of the set of points of convergence of a power series, whose radius is defined as the series' convergence radius.

Step 3: Find the Region of convergence.

Use the ratio test. 

$\begin{array}{l}{\mathrm{\rho }}_{\mathrm{n}}=\frac{{\mathrm{a}}_{\mathrm{n}+1}}{{\mathrm{a}}_{\mathrm{n}}}\\ {\mathrm{\rho }}_{\mathrm{n}}=\left|\begin{array}{c}\frac{{\mathrm{z}}^{\mathrm{n}+1}}{{\mathrm{z}}^{\mathrm{n}}}\end{array}\right|\end{array}$ 

 

Calculate the value of $\mathrm{\rho }$, i.e.,

$\begin{array}{l}\mathrm{p}=\underset{\mathrm{n}\to \infty }{\lim }{\mathrm{\rho }}_{\mathrm{n}}\\ {\mathrm{\rho }}_{\mathrm{n}}=\underset{\mathrm{n}\to \infty }{\lim }\left|\frac{{\mathrm{z}}^{\mathrm{n}+1}}{{\mathrm{z}}^{\mathrm{n}}}\right|\\ {\mathrm{\rho }}_{\mathrm{n}}=\left|\mathrm{z}\right|\end{array}$ 

 

${\mathrm{S}}_{\mathrm{n}}$ is convergent for $\mathrm{\rho }<1$, i.e., $\left|z\right|<1$.

 

Hence, the region of convergence is $\left|z\right|<1$.