1–10. Choosing convergence tests Identify a convergence test for each series. If necessary, explain how to simplify or rewrite the series before applying the convergence test. You do not need to carry out the convergence test.
∑ (from k = 3 to ∞) (2k²) / (k² − k − 2)

Verified step by step guidance

1. First, examine the general term of the series: \(\frac{2k^{2}}{k^{2} - k - 2}\). To understand its behavior, try to simplify the denominator by factoring it.
2. Factor the quadratic expression in the denominator: \(k^{2} - k - 2 = (k - 2)(k + 1)\). So the term becomes \(\frac{2k^{2}}{(k - 2)(k + 1)}\).
3. Next, analyze the behavior of the term for large \(k\). Since the numerator and denominator are both degree 2 polynomials, consider the limit of the term as \(k \to \infty\) to compare it with a simpler series.
4. Because the degrees of numerator and denominator are the same, the term behaves like a constant times \(\frac{k^{2}}{k^{2}}\), which simplifies to a constant. This suggests the terms do not approach zero, which is a key condition for convergence.
5. Based on this, the Divergence Test (also called the Test for Divergence) is the appropriate convergence test to apply first, since if the terms do not approach zero, the series cannot converge.

Key Concepts

Series and Convergence

A series is the sum of the terms of a sequence. Understanding whether a series converges (approaches a finite limit) or diverges is fundamental in calculus. Convergence tests help determine this behavior without explicitly finding the sum.

Simplifying Rational Expressions

Before applying convergence tests, it is often helpful to simplify the general term of the series. Factoring the denominator or numerator can reveal cancellations or simpler forms, making it easier to identify the dominant behavior of terms as k approaches infinity.

Convergence Tests for Series

Common tests include the Comparison Test, Limit Comparison Test, and the Divergence Test. Choosing the right test depends on the form of the series terms; for rational functions, comparing to a simpler p-series or using the Limit Comparison Test is often effective.