11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞)cos(1 / k⁹)

Verified step by step guidance

1. First, write down the general term of the series: \(a_k = \cos\left(\frac{1}{k^9}\right)\).
2. Recall that for a series \(\sum a_k\) to converge, the terms \(a_k\) must approach zero as \(k\) approaches infinity. So, evaluate \(\lim_{k \to \infty} a_k = \lim_{k \to \infty} \cos\left(\frac{1}{k^9}\right)\).
3. Since \(\frac{1}{k^9} \to 0\) as \(k \to \infty\), use the continuity of cosine to find \(\lim_{k \to \infty} \cos\left(\frac{1}{k^9}\right) = \cos(0) = 1\).
4. Because the terms \(a_k\) do not approach zero (they approach 1), the necessary condition for convergence of the series is not met. Therefore, the series \(\sum_{k=1}^\infty \cos\left(\frac{1}{k^9}\right)\) diverges.
5. In conclusion, no further convergence tests are needed since the terms do not tend to zero, and the series diverges by the Test for Divergence (also called the nth-term test).

Key Concepts

Convergence of Infinite Series

An infinite series converges if the sequence of its partial sums approaches a finite limit. Understanding convergence is essential to determine whether the sum of infinitely many terms results in a finite value or diverges to infinity or oscillates.

Limit Comparison and Behavior of Terms

For a series to converge, its terms must approach zero as k approaches infinity. Analyzing the limit of the general term, such as cos(1/k⁹), helps determine if the terms tend to zero or a nonzero value, which is a necessary condition for convergence.

Convergence Tests for Series

Various tests like the Comparison Test, Limit Comparison Test, and the Divergence Test help determine series convergence. Applying these tests to the given series involves comparing it to known convergent or divergent series or examining the limit of terms to justify convergence or divergence.