Q1P - Mathematical Methods in Physical Sciences

Question

Prove that an absolutely convergent series of complex numbers converges. This means to prove that $\mathbf{\sum }\mathbf{(}{\mathbf{a}}_{\mathbf{n}}\mathbf{+}{\mathbf{ib}}_{\mathbf{n}}\mathbf{)}\mathbf{ }$converges ($\mathbf{ }{\mathbf{a}}_{\mathbf{n}}$and ${\mathbf{b}}_{\mathbf{n}}$real) if $\mathbf{\sum }\sqrt{{{\mathbf{a}}_{\mathbf{n}}}^{\mathbf{2}}\mathbf{+}{{\mathbf{b}}_{\mathbf{n}}}^{\mathbf{2}}}\mathbf{ }$converges. Hint: Convergence of $\mathbf{\sum }\mathbf{(}{\mathbf{a}}_{\mathbf{n}}\mathbf{+}{\mathbf{ib}}_{\mathbf{n}}\mathbf{)}$means that $\mathbf{\sum }{\mathbf{a}}_{\mathbf{n}}\mathbf{ }$and $\mathbf{\sum }{\mathbf{b}}_{\mathbf{n}}$ both converge. Compare $\mathbf{\sum }\mathbf{\left|}{\mathbf{a}}_{\mathbf{n}}\mathbf{\right|}$ and$\mathbf{ }\mathbf{\sum }\mathbf{\left|}{\mathbf{b}}_{\mathbf{n}}\mathbf{\right|}$ with $\mathbf{\sum }\sqrt{{{\mathbf{a}}_{\mathbf{n}}}^{\mathbf{2}}\mathbf{+}{{\mathbf{b}}_{\mathbf{n}}}^{\mathbf{2}}}$  , and use Problem 7.9 of Chapter 1 .

Step-by-Step Solution

Answer

The statement has been proven.

Step 1: Given Information

Two series ${\mathrm{a}}_{\mathrm{n}},{\mathrm{b}}_{\mathrm{n}}$ .

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: Prove the statement

Let S be the series:

$$\begin{gathered} S=\underset{ }{\sum C_n} \\ =\underset{ }{\sum a_n+i{b}_n} \end{gathered}$$ 

The magnitude of $C_n$is $C_n=\sqrt{a_n^2+b_n^2}$ .

  $$\begin{gathered} {\mathrm{a}}_{\mathrm{n}}={\mathrm{C}}_{\mathrm{n}} \mathrm{cos\theta } \\ =\sqrt{{\mathrm{a}}_{\mathrm{n}}^2+{\mathrm{b}}_{\mathrm{n}}^2} \mathrm{cos\theta } \\ \left|{\mathrm{a}}_{\mathrm{n}}\right|<\left|{\mathrm{C}}_{\mathrm{n}}\right| \end{gathered}$$

 

 Hence $a_n$  is convergent.

 $$\begin{gathered} b_{\mathrm{n}}={\mathrm{C}}_{\mathrm{n}} \sin \theta \\ =\sqrt{{\mathrm{a}}_{\mathrm{n}}^2+{\mathrm{b}}_{\mathrm{n}}^2} \sin \theta \\ \left|b_{\mathrm{n}}\right|<\left|{\mathrm{C}}_{\mathrm{n}}\right| \end{gathered}$$

 

Hence $b_n$ is convergent.

Thus, an absolutely convergent series of complex numbers converges. The statement has been proven.