Question

Test for convergence: $$\sum_{n=2}^{\infty} \frac{2 n^{3}}{n^{4}-2}$$

Short answer

The series diverges by the Comparison Test.

Step-by-step solution

Identify the Series

The given series is \(\sum_{n=2}^{\infty} \frac{2 n^{3}}{n^{4}-2}\). We need to determine if this series converges.

Simplify the General Term

Simplify the general term \(a_n = \frac{2 n^{3}}{n^{4} - 2}\). For large values of \(n\), the term \(n^{4} - 2\) is dominated by \(n^{4}\), so the general term approximates to \(a_n \approx \frac{2 n^{3}}{n^{4}} = \frac{2}{n}\).

Use the Comparison Test

Use the Comparison Test to compare \(a_n\) with the harmonic series term \(\frac{2}{n}\). Note that \(\sum_{n=1}^{\infty} \frac{1}{n}\) (the harmonic series) diverges. We need to confirm if \(\sum_{n=2}^{\infty} \frac{2n^3}{n^4-2}\) has similar behavior.

Establishing the Comparison

Since \(\frac{2 n^{3}}{n^{4} - 2}\) is greater than \(\frac{2}{n+1}\) for \(n \ge 2\), and \(\sum_{n=2}^{\infty} \frac{2}{n}\) diverges, by the Comparison Test, the given series \(\sum_{n=2}^{\infty} \frac{2 n^{3}}{n^{4} - 2}\) also diverges.

Key concepts

comparison test

The Comparison Test is a handy tool to determine the convergence or divergence of a series by comparing it to a second series whose behavior is already known. To use it effectively, you follow these steps:

• Identify the given series and simplify the general term if possible.
• Find a second series whose convergence or divergence is known.
• Compare the terms of the two series to decide if the original series converges or diverges.

The essential principle is that if your series has terms that are consistently larger than or comparable to a divergent series, your series will diverge as well. Conversely, if your series has terms that are consistently smaller than or comparable to a convergent series, your series will converge. In the given example, we compared \(\frac{2n^3}{n^4-2}\) with \(\frac{2}{n}\). Because we know that \(\frac{2}{n}\) (the harmonic series) diverges, and \(\frac{2n^3}{n^4-2}\) is larger than \(\frac{2}{n+1}\) for all .

harmonic series

The harmonic series is one of the fundamental examples in the study of infinite series. It is given by \(\textstyle\sum_{n=1}^{\infty} \frac{1}{n}\) and is known for its divergence. Despite its terms becoming smaller and smaller, the sum of the series grows without bound. Here are some key points about the harmonic series:

• It diverges because it can be compared to the integral \(\textstyle\begin{align*}\textstyle\int_1^ x \frac{1}{t} dt\end{align*}\)
• The divergence can be understood through the Comparison Test by splitting the harmonic series into manageable partial sums.

Understanding the properties of the harmonic series is crucial because it sets a precedent for evaluating more complex series like the one in the given exercise.

approximation of terms

Approximating terms is a common technique used to simplify the analysis of complex series. The idea is to focus on the behavior of the series as \ grows very large. Here’s how to approach it:

• First, identify the leading term in the numerator and the denominator of the general term of the series.
• For large values of \, ignore the smaller terms compared to the dominant term.
• Use this approximate term to compare with known series for convergence or divergence analysis.

In this exercise, the term \(\textstyle\frac{2n^3}{n^4-2}\) simplifies to \(\textstyle\frac{2}{n}\) for large values of \. This approximation made it easier to compare our series with the harmonic series and thereby conclude its divergence. Approximations thus provide a streamlined method to handle otherwise complicated expressions.