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

Verified step by step guidance

1. Identify the series given: \( \sum_{k=1}^{\infty} \sin\left(\frac{1}{k^9}\right) \). We want to determine if this series converges.
2. Recall that for very small angles \( x \), \( \sin x \approx x \). Since \( \frac{1}{k^9} \) becomes very small as \( k \to \infty \), we can approximate \( \sin\left(\frac{1}{k^9}\right) \approx \frac{1}{k^9} \) for large \( k \).
3. Compare the given series to the p-series \( \sum_{k=1}^{\infty} \frac{1}{k^p} \) where \( p = 9 \). We know that a p-series converges if \( p > 1 \). Since 9 is much greater than 1, the p-series \( \sum \frac{1}{k^9} \) converges.
4. Use the Limit Comparison Test to justify convergence: compute \( \lim_{k \to \infty} \frac{\sin\left(\frac{1}{k^9}\right)}{\frac{1}{k^9}} \). If this limit is a finite nonzero number, then both series behave similarly in terms of convergence.
5. Since the limit comparison test shows the given series behaves like a convergent p-series, conclude that \( \sum_{k=1}^{\infty} \sin\left(\frac{1}{k^9}\right) \) converges.

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.

Comparison Test for Series

The comparison test involves comparing the given series to a known benchmark series. If the terms of the given series are smaller than those of a convergent series, it also converges; if larger than a divergent series, it diverges. This test helps analyze series with complicated terms.

Behavior of sin(x) for Small x

For values of x close to zero, sin(x) is approximately equal to x. This approximation allows simplification of terms like sin(1/k⁹) to 1/k⁹ for large k, facilitating the use of p-series tests to determine convergence.