Question

11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.1 / (2·3) + 1 / (4·5) + 1 / (6·7) + 1 / (8·9) + ⋯

Accepted Answer

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 $$ a_n = \frac{1}{(2n)(2n+1)} . $$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. 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. 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 $$N 	o \infty. $$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.

11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
1 / (2·3) + 1 / (4·5) + 1 / (6·7) + 1 / (8·9) + ⋯

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Key Concepts

Convergence of Infinite Series

Comparison Test

General Term Analysis and Simplification