Question

Test each of the following series for convergence.

$\mathbf{\sum }\frac{\mathbf{1}}{{\mathbf{n}}^{\mathbf{2}}}\mathbf{+}\frac{\mathbf{i}}{\mathbf{n}}$

Short answer

The series is divergent.

Step-by-step solution

Given Information

The series is $\sum \frac{1}{{\mathrm{n}}^2}+\frac{\mathrm{i}}{\mathrm{n}}$.

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.

Test the convergence.

The series is $\sum \frac{1}{{\mathrm{n}}^2}+\frac{\mathrm{i}}{\mathrm{n}}=\sum \frac{1}{{\mathrm{n}}^2}+\sum \frac{\mathrm{i}}{\mathrm{n}}.$.

The given series is the sum of convergent and divergent series hence the series is divergent.