Q3P - Mathematical Methods in Physical Sciences

Question

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

$\mathbf{1}\mathbf{-}\frac{{\mathbf{z}}^{\mathbf{2}}}{\mathbf{3}\mathbf{!}}\mathbf{+}\frac{{\mathbf{z}}^{\mathbf{4}}}{\mathbf{5}\mathbf{!}}\mathbf{-}\mathbf{·}\mathbf{·}\mathbf{·}$

Step-by-Step Solution

Answer

The entire complex plane is the region of convergence.

Step 1: Given Information.

The given power series, i.e.,  ${\mathrm{S}}_{\mathrm{n}}=1-\frac{{\mathrm{z}}^2}{3!}+\frac{{\mathrm{z}}^4}{5!}-···$

Step 2: Definition of Disc 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 general term of the series.

Use the series to find the general term.

$$\begin{gathered} {\mathrm{S}}_{\mathrm{n}}=1-\frac{{\mathrm{z}}^2}{3!}+\frac{{\mathrm{z}}^4}{5!}-··· ···\left(1\right) \\ {\mathrm{S}}_{\mathrm{n}}=\underset{ }{\sum {\displaystyle \frac{{\left(-1\right)}^{\mathrm{n}}{\mathrm{z}}^{2\mathrm{n}}}{\left(\mathrm{n}+1\right)}}} ···\left(2\right) \end{gathered}$$ 

Step 4: Find the Region of convergence.

Use the ratio test. 

 $$\begin{gathered} {\mathrm{\rho }}_{\mathrm{n}}=\frac{{\mathrm{a}}_{\mathrm{n}+1}}{{\mathrm{a}}_{\mathrm{n}}} \\ {\mathrm{\rho }}_{\mathrm{n}}=\left|\frac{{\displaystyle \frac{{\left(-1\right)}^{\mathrm{n}+1}{\mathrm{z}}^{2\left(\mathrm{n}+1\right)}}{\left(\mathrm{n}+2\right)!}}}{{\displaystyle \frac{{\left(-1\right)}^{\mathrm{n}}{\mathrm{z}}^{2\mathrm{n}}}{\left(\mathrm{n}+1\right)!}}}\right| \\ =\left|-\frac{{\mathrm{z}}^2}{\mathrm{n}+2}\right| \end{gathered}$$

 

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

 $\begin{array}{l}\mathrm{\rho }=\underset{\mathrm{n}\to \infty }{\lim }{\mathrm{\rho }}_{\mathrm{n}}\\ =\underset{\mathrm{n}\to \infty }{\lim }\left|-\frac{{\mathrm{z}}^2}{\mathrm{n}+2}\right|\\ =0\end{array}$

 

${\mathrm{S}}_{\mathrm{n}}$ is convergent for all values of z.

 

Hence, the entire complex plane is the region of convergence.