Q8P - Mathematical Methods in Physical Sciences

Question

Test each of the following series for convergence.

$\mathbf{ }\mathbf{\sum }_{\mathbf{ }}{\mathit{e}}^{\mathbf{i}\mathbf{n}\frac{\mathbf{\pi }}{\mathbf{6}}}$

Step-by-Step Solution

Answer

The series is divergent.

Step 1: Given Information

The series is $\underset{ }{\sum e^{\frac{in\mathrm{\pi }}{6}}}$.

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 $\underset{ }{\sum e^{\frac{in\mathrm{\pi }}{6}}}$.

$$\begin{gathered} A_n=e^{in\mathrm{\pi }/6} \\ {\mathrm{A}}_{\mathrm{n}+1}={\mathrm{e}}^{\mathrm{i}(\mathrm{n}+1)\mathrm{\pi }/6} \end{gathered}$$ 

 

 Find the ratio $\left|\frac{{\mathrm{A}}_{\mathrm{n}+1}}{{\mathrm{A}}_{\mathrm{n}}}\right|$.

$$\begin{gathered} {\mathrm{p}}_{\mathrm{n}}=\left|\frac{{\mathrm{A}}_{\mathrm{n}+1}}{{\mathrm{A}}_{\mathrm{n}}}\right| \\ =\frac{{\mathrm{e}}^{\mathrm{i}(\mathrm{n}+1)\mathrm{\pi }/6}}{{\mathrm{e}}^{\mathrm{i}\left(\mathrm{n}\right)\mathrm{\pi }/6}} \\ ={\mathrm{e}}^{\mathrm{i\pi }/6} \end{gathered}$$ 

 

Find the limit $\mathrm{p}=\underset{\mathrm{n}\to \infty }{\lim }\frac{{\mathrm{A}}_{\mathrm{n}+1}}{{\mathrm{A}}_{\mathrm{n}}}$ 

$$\begin{gathered} \mathrm{p}=\left|{\mathrm{e}}^{\mathrm{i\pi }/6}\right| \\ =\left| \cos \left(\mathrm{\pi }/6\right)+\mathrm{i} \sin \left(\mathrm{\pi }/6\right)\right| \\ =1 \end{gathered}$$ 

 

Test fails.

 

Find the sum of the series.

$$\begin{gathered} \mathrm{S}={\int }_1^{\infty }{\mathrm{e}}^{\mathrm{in\pi }/6}\mathrm{dn} \\ =\frac{6}{\mathrm{i\pi }}{\left[{\mathrm{e}}^{\mathrm{in\pi }/6}\right]}_1^{\infty } \end{gathered}$$ 

 

The sum is not defined.

 

Hence the series is divergent.