Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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**Series Analysis for Convergence or Divergence** **Problem Statement:** Evaluate the convergence or divergence of the following infinite series. If a series is convergent, determine its sum. Justify the answer by naming the relevant property and demonstrating that your series meets the necessary conditions. **Series to Analyze:** (A) \(\sum_{k=1}^{\infty} (2)^k (3)^{-k}\) (B) \(\sum_{k=1}^{\infty} \left(\frac{1}{k}\right)\) (C) \(\sum_{k=1}^{\infty} \left(\frac{4^k}{5^k}\right)\) **Explanation:** 1. **Series (A):** The terms are expressed as \((2)^k(3)^{-k}\), which simplifies to \(\left(\frac{2}{3}\right)^k\). This is a geometric series with each term having a common ratio \(r = \frac{2}{3}\). 2. **Series (B):** The terms are expressed as \(\frac{1}{k}\). This series is the harmonic series, which is known for diverging as \(k\) approaches infinity. 3. **Series (C):** The terms are expressed as \(\frac{4^k}{5^k}\), which simplifies to \(\left(\frac{4}{5}\right)^k\). This is also a geometric series with a common ratio \(r = \frac{4}{5}\). **Instructions:** Determine whether each series converges or diverges by applying appropriate convergence tests, such as the geometric series test or the harmonic series test. For convergent series, calculate the sum using relevant formulas or properties.
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Transcribed Image Text:**Series Analysis for Convergence or Divergence** **Problem Statement:** Evaluate the convergence or divergence of the following infinite series. If a series is convergent, determine its sum. Justify the answer by naming the relevant property and demonstrating that your series meets the necessary conditions. **Series to Analyze:** (A) \(\sum_{k=1}^{\infty} (2)^k (3)^{-k}\) (B) \(\sum_{k=1}^{\infty} \left(\frac{1}{k}\right)\) (C) \(\sum_{k=1}^{\infty} \left(\frac{4^k}{5^k}\right)\) **Explanation:** 1. **Series (A):** The terms are expressed as \((2)^k(3)^{-k}\), which simplifies to \(\left(\frac{2}{3}\right)^k\). This is a geometric series with each term having a common ratio \(r = \frac{2}{3}\). 2. **Series (B):** The terms are expressed as \(\frac{1}{k}\). This series is the harmonic series, which is known for diverging as \(k\) approaches infinity. 3. **Series (C):** The terms are expressed as \(\frac{4^k}{5^k}\), which simplifies to \(\left(\frac{4}{5}\right)^k\). This is also a geometric series with a common ratio \(r = \frac{4}{5}\). **Instructions:** Determine whether each series converges or diverges by applying appropriate convergence tests, such as the geometric series test or the harmonic series test. For convergent series, calculate the sum using relevant formulas or properties.
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