Question

Test each of the following series for convergence.

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

Short answer

The series is divergent.

Step-by-step solution

Given Information

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

Definition of the Convergent series

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

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

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 $\sum (1+i{)}^n$.

$$\begin{gathered} A_n={\left(1+i\right)}^n \\ {A}_{n+1}={\left(1+i\right)}^{n+1} \end{gathered}$$

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

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

$\rho >1$, Hence the series is divergent.