27–37. Evaluating series Evaluate the following infinite series or state that the series diverges.
∑ (from k = 1 to ∞)ln((2k + 1) / (2k − 1))

Verified step by step guidance

1. Start by writing out the general term of the series: \( a_k = \ln\left(\frac{2k + 1}{2k - 1}\right) \).
2. Use the logarithm property \( \ln\left(\frac{A}{B}\right) = \ln(A) - \ln(B) \) to rewrite the term as \( a_k = \ln(2k + 1) - \ln(2k - 1) \).
3. Express the partial sum \( S_n = \sum_{k=1}^n a_k \) by substituting the expanded terms: \( S_n = \sum_{k=1}^n \left( \ln(2k + 1) - \ln(2k - 1) \right) \).
4. Rewrite the partial sum as \( S_n = \left( \ln 3 - \ln 1 \right) + \left( \ln 5 - \ln 3 \right) + \left( \ln 7 - \ln 5 \right) + \cdots + \left( \ln(2n + 1) - \ln(2n - 1) \right) \) and observe the telescoping pattern where most terms cancel out.
5. Simplify the telescoping sum to \( S_n = \ln(2n + 1) - \ln 1 \), then analyze the limit \( \lim_{n \to \infty} S_n = \lim_{n \to \infty} \ln(2n + 1) \) to determine whether the series converges or diverges.

Key Concepts

Infinite Series and Convergence

An infinite series is the sum of infinitely many terms. To evaluate such a series, it is crucial to determine whether it converges (approaches a finite limit) or diverges (grows without bound or oscillates). Convergence tests help decide if the sum exists.

Properties of Logarithms

Logarithmic properties, such as ln(a/b) = ln(a) - ln(b), allow simplification of terms in the series. Applying these properties can transform the series into a telescoping form or other manageable expressions, facilitating evaluation.

Telescoping Series

A telescoping series is one where many terms cancel out when the series is expanded, leaving only a few terms to sum. Recognizing telescoping behavior is key to simplifying and finding the sum of certain infinite series involving differences of logarithms.