11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
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Verified step by step guidance

1. Identify the general term of the series. Notice the pattern in the denominators: the first term is \( \frac{1}{2 \cdot 3} \), the second is \( \frac{1}{4 \cdot 5} \), the third is \( \frac{1}{6 \cdot 7} \), and so on. We can express the \(n\)-th term as \( a_n = \frac{1}{(2n)(2n+1)} \).
2. Simplify the general term using partial fraction decomposition. Write \( \frac{1}{(2n)(2n+1)} \) as \( \frac{A}{2n} + \frac{B}{2n+1} \) and solve for constants \(A\) and \(B\). This will help in understanding the behavior of the series.
3. After finding the partial fractions, rewrite the series as a telescoping series if possible. Telescoping series have terms that cancel out sequentially, making it easier to analyze convergence.
4. Evaluate the partial sums of the series using the telescoping form. Write the sum of the first \(N\) terms and simplify to see if the sum approaches a finite limit as \(N \to \infty\).
5. Conclude about the convergence of the series based on the behavior of the partial sums. If the partial sums approach a finite number, the series converges; otherwise, it diverges.

Key Concepts

Convergence of Infinite Series

An infinite series converges if the sum of its terms approaches a finite limit as the number of terms increases indefinitely. Understanding convergence is essential to determine whether adding infinitely many terms results in a finite value or diverges to infinity.

Comparison Test

The Comparison Test involves comparing the given series to a known benchmark series with established convergence behavior. 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.

General Term Analysis and Simplification

Expressing the general term of the series in a simplified form helps identify its behavior and apply convergence tests effectively. For example, rewriting terms like 1/(2n(2n+1)) can reveal similarities to p-series or telescoping series.