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 = 10 to ∞) 1 / (k − 9)⁵

Verified step by step guidance

1. First, rewrite the series to simplify the expression inside the summation. Notice that the term is \( \frac{1}{(k - 9)^5} \). Let \( n = k - 9 \), so when \( k = 10 \), \( n = 1 \). Thus, the series becomes \( \sum_{n=1}^{\infty} \frac{1}{n^5} \).
2. Recognize that the rewritten series is a p-series of the form \( \sum_{n=1}^{\infty} \frac{1}{n^p} \) where \( p = 5 \).
3. Recall the p-series convergence test: a p-series converges if and only if \( p > 1 \). Since \( p = 5 > 1 \), this series converges.
4. Therefore, the appropriate convergence test to identify this series' behavior is the p-series test.
5. No further simplification is necessary because the series is already in a standard form suitable for applying the p-series test.

Key Concepts

Infinite Series and Convergence

An infinite series is the sum of infinitely many terms. Understanding whether such a series converges (approaches a finite value) or diverges is fundamental. Convergence depends on the behavior of the terms as the index grows large.

p-Series Test

The p-series test applies to series of the form ∑ 1/n^p. It states that the series converges if p > 1 and diverges otherwise. Recognizing a series as a p-series or rewriting it into this form helps quickly determine convergence.

Index Shifting and Simplification

Rewriting or shifting the index of summation can simplify a series to a more recognizable form. For example, changing variables to start the index at 1 can help identify the series type and apply appropriate convergence tests.