Question

Test each of the following series for convergence. $$\sum\left(\frac{1}{n^{2}}+\frac{i}{n}\right)$$

Short answer

The series \(\sum \left(\frac{1}{n^2} + \frac{i}{n}\right)\) diverges.

Step-by-step solution

Identify the Series Components

The series given is \sum \left(\frac{1}{n^{2}}+\frac{i}{n}\right). It consists of two separate series: \(\sum \frac{1}{n^{2}}\) and \(\sum \frac{i}{n}\). Test each part individually for convergence.

Test \(\sum \frac{1}{n^2}\) for Convergence using the p-Series Test

Recognize that \(\sum \frac{1}{n^2}\) is a p-series with \(p = 2\). A p-series converges if \(p > 1\). Since \(2 > 1\), \(\sum \frac{1}{n^2}\) converges.

Test \(\sum \frac{i}{n}\) for Convergence

Note that \(i\) is a constant (the imaginary unit), and can be factored out of the series. Thus, \(\sum \frac{i}{n} = i \sum \frac{1}{n}\). The series \(\sum \frac{1}{n}\) (the harmonic series) is known to diverge. Therefore, \(i \sum \frac{1}{n}\) also diverges.

Combine the Results

Since \(\sum \frac{1}{n^2}\) converges but \(\sum \frac{i}{n}\) diverges, the given series \(\sum \left(\frac{1}{n^2} + \frac{i}{n}\right)\) diverges because the divergence of any part of a sum causes the whole series to diverge.

Key concepts

p-series test

The p-series test is a common method for determining the convergence or divergence of a series. It evaluates series of the form \(\frac{1}{n^p}\). The test states that if \p > 1\, the series converges. Otherwise, if \p ≤ 1\, the series diverges.
In our exercise, we had to test the series \(\frac{1}{n^2}\). Here, \p = 2\, and since \2 > 1\, the p-series test confirms that this part of the series converges.

By understanding this concept, we can quickly identify that any series where \p > 1\ will indeed converge, saving time in evaluating complex problems.

imaginary unit

In mathematics, the imaginary unit, denoted as \i\, is defined as the square root of -1. It is used to extend the real number system to the complex number system, which is crucial in various fields of science and engineering.
When dealing with series involving imaginary numbers, like \(\frac{i}{n}\), it is important to treat \i\ as a constant factor and consider the remaining part of the series independently. In our example, \i \is factored out, transforming the series into \i \sum \frac{1}{n}\.

Although \i\ makes the series complex, understanding its properties allows us to apply familiar tests (like the harmonic series test) effectively.

harmonic series

A harmonic series is one of the most fundamental series in mathematics. It is defined as \ \sum \frac{1}{n}\. Unlike the p-series with \p > 1\, the harmonic series diverges, meaning that it grows indefinitely as n increases.

In our example, the presence of the harmonic series \(\frac{1}{n}\) compounded with \i\ indicates divergence regardless of the imaginary unit. Because any divergent component in a combined series results in the entire series diverging, the harmonic series dictates the behavior of the entire sum.

Understanding harmonic series is fundamental in recognizing divergence patterns, which has applications in evaluating complex and real series in advanced mathematical contexts.