Q31MP - Mathematical Methods in Physical Sciences

Question

Show that if a sequence of complex numbers tends to zero, then the sequence of absolute values tends to zero too, and vice versa. Hint${\mathbf{a}}_{\mathbf{n}}\mathbf{+}{\mathbf{ib}}_{\mathbf{n}}\mathbf{\to }\mathbf{0}$ means $\mathbf{ }\mathbf{ }{\mathbf{a}}_{\mathbf{n}}\mathbf{\to }\mathbf{0}\mathbf{ }$ and${\mathbf{b}}_{\mathbf{n}}\mathbf{\to }\mathbf{0}\mathbf{.}$ 

Step-by-Step Solution

Answer

It has been proved that ${\mathrm{a}}_{\mathrm{n}}+{\mathrm{ib}}_{\mathrm{n}}\to 0$ means 

Step 1: Given Information.

It has been given that the complex numbers tends to zero.

Step 2: Meaning of rectangular form.

Represent the complex number in rectangular form means writing the given complex number in the form of $\mathbf{x}\mathbf{+}\mathbf{iy}\mathbf{ }$in which x is the real part and y is the imaginary part.

Step 3: Solve the series.

Assume a sequence of complex number.

$$\begin{gathered} S=\sum C_n \\ S=C_1+C_2+C_3+\cdots \end{gathered}$$

 

A complex number has a real and an imaginary part.

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

 

If tends to be zero thentends to be zero and this is only possible when must be zero.