Q4P - Mathematical Methods in Physical Sciences

Question

Test each of the following series for convergence.

$\mathbf{\sum }{\left(\frac{1-i}{1+i}\right)}^{\mathbf{n}}$

Step-by-Step Solution

Answer

The series is divergent.

Step 1: Given Information

The series is $\sum {\left(\frac{1-\mathrm{i}}{1+\mathrm{i}}\right)}^n$.

Step 2: Definition of the Convergent series.

A series is said to be convergent if the terms of a series get close to zero when the number of terms moves towards infinity.

Step 3: Test the convergence.

The series is $\sum {\left(\frac{1-\mathrm{i}}{1+\mathrm{i}}\right)}^n$ .

 

Find the limit $\underset{\mathrm{n}\to \infty }{\lim }\left|\frac{{\mathrm{A}}_{\mathrm{n}+1}}{{\mathrm{A}}_{\mathrm{n}}}\right|$.

$$\begin{gathered} \mathrm{\rho }=\underset{\mathrm{n}\to \infty }{\lim }\left|\frac{{\mathrm{A}}_{\mathrm{n}+1}}{{\mathrm{A}}_{\mathrm{n}}}\right| \\ =\underset{\mathrm{n}\to \infty }{\lim }\left|\frac{{\left[\left(1-\mathrm{i}\right)/\left(1+\mathrm{i}\right)\right]}^{\mathrm{n}+1}}{{\left[\left(1-\mathrm{i}\right)/\left(1+\mathrm{i}\right)\right]}^{\mathrm{n}}}\right| \\ =\underset{\mathrm{n}\to \infty }{\lim }\left|\frac{1-\mathrm{i}}{1+\mathrm{i}}\right| \\ =1 \end{gathered}$$ 

 

Test fails.

 

Find the sum of the series.

$$\begin{gathered} \mathrm{a}=\frac{1-\mathrm{i}}{1+\mathrm{i}} \\ \mathrm{S}={\int }_0^{\infty }{\mathrm{a}}^{\mathrm{n}}\mathrm{dn} \\ ={\left[\frac{{\mathrm{a}}^{\mathrm{n}}}{\mathrm{In}\left(\mathrm{a}\right)}\right]}_{-1}^{\infty } \\ =\infty \end{gathered}$$

The sum is not finite.

 

Hence the series is divergent.