Q5P - Mathematical Methods in Physical Sciences

Question

Test each of the following series for convergence.

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

Step-by-Step Solution

Answer

The series is divergent.

Step 1: Given Information

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

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: 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.