Question

Test each of the following series for convergence.

$\mathbf{\sum }\mathbf{1}\mathbf{/}\mathbf{(}\mathbf{1}\mathbf{+}\mathit{i}{\mathbf{)}}^{\mathbf{n}}$

Short answer

The series is convergent.

Step-by-step solution

Given Information

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

Definition of the Convergent series

We can check if the series converges or not by calculating as:

$\mathbf{\rho }\mathbf{=}\underset{\mathbf{n}\mathbf{\to }\mathbf{\infty }}{\mathbf{lim}}\mathbf{\left|}\frac{{\mathbf{A}}_{\mathbf{n}\mathbf{+}\mathbf{1}}}{{\mathbf{A}}_{\mathbf{n}}}\mathbf{\right|}$

If $\mathit{\rho }\mathbf{<}\mathbf{1}$, then the series converges.

If $\mathit{\rho }\mathbf{>}\mathbf{1}$, then the series diverges.

Test the convergence

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

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

$$\begin{gathered} \mathrm{\rho }=\underset{\mathrm{n}\to \infty }{\lim }\left|\frac{{\mathrm{A}}_{\mathrm{n}+1}}{{\mathrm{A}}_{\mathrm{n}}}\right| \\ =\underset{\mathrm{n}\to \infty }{\lim }\left|\frac{{\displaystyle \frac{1}{{\left(1+i\right)}^{n+2}}}}{{\displaystyle \frac{1}{{\left(1+i\right)}^n}}}\right| \\ =\underset{\mathrm{n}\to \infty }{\lim }\left|\frac{1}{1+i}\right| \\ =\left|\frac{1}{\sqrt{2}}\right| \end{gathered}$$

$\rho <1$, Hence the series is convergent.