Q13P - Mathematical Methods in Physical Sciences

Question

Test each of the following series for convergence.

$\mathbf{\sum }_{\mathbf{ }}\frac{{\mathbf{(}\mathbf{1}\mathbf{+}\mathit{i}\mathbf{)}}^{\mathbf{n}}}{{\mathbf{(}\mathbf{2}\mathbf{=}\mathbf{i}\mathbf{)}}^{\mathbf{n}}}$

Step-by-Step Solution

Answer

The series converges, i.e., $\rho <1$ .

Step 1: Given Information.

The given series, i.e., $\sum _{ }\frac{{(1+i)}^{\mathrm{n}}}{{(2=\mathrm{i})}^{\mathrm{n}}}$ 

Step 2: Definition of Convergent and Divergent series.

A convergent series is one in which the partial sums all gravitate to the same finite number, also known as a limit. Divergent refers to any series that is not converging.

Step 3: Calculate the value of &#961; n .

Find the value of  $a_n$ and  $a_{n+1}$ 

 $$\begin{gathered} {\mathrm{a}}_{\mathrm{n}}={\left(\frac{1+\mathrm{i}}{2-\mathrm{i}}\right)}^n ...\left(1\right) \\ {\mathrm{a}}_{\mathrm{n}+1}={\left(\frac{1+\mathrm{i}}{2-\mathrm{i}}\right)}^{n+1} ...\left(2\right) \end{gathered}$$

 

If  $\rho <1$ series converges, if $\rho >1$ then diverges.

 

$\mathrm{\rho }=\underset{\mathrm{n}\to \infty }{\lim }{\mathrm{\rho }}_{\mathrm{n}}$

Step 4: Test the series for convergence.

Use the ratio test. 

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

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

$$\begin{gathered} \mathrm{\rho }=\underset{\mathrm{n}\to \infty }{\lim }\left|{\mathrm{\rho }}_{\mathrm{n}}\right| \\ =\underset{\mathrm{n}\to \infty }{\lim }\left|\frac{1+\mathrm{i}}{2-\mathrm{i}}\right| \\ =\left|0.2+0.6\mathrm{i}\right| \\ =0.632 \end{gathered}$$

Hence, the series converges, i.e.,  $\mathrm{\rho }<1$.