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

∑ (from j = 1 to ∞)cot(–1 / j) / 2ʲ

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Rewrite the general term of the series to better understand its behavior: the series is \( \sum_{j=1}^\infty \frac{\cot\left(-\frac{1}{j}\right)}{2^j} \). Note that \( 2^j \) grows exponentially, which suggests the denominator grows quickly.

Use the odd function property of cotangent: since \( \cot(-x) = -\cot(x) \), rewrite the term as \( \frac{-\cot\left(\frac{1}{j}\right)}{2^j} = -\frac{\cot\left(\frac{1}{j}\right)}{2^j} \). This helps simplify the analysis.

Analyze the behavior of \( \cot\left(\frac{1}{j}\right) \) as \( j \to \infty \). Since \( \frac{1}{j} \to 0 \), use the approximation \( \cot x \approx \frac{1}{x} \) for small \( x \). Thus, \( \cot\left(\frac{1}{j}\right) \approx j \).

Substitute the approximation into the term: \( -\frac{\cot\left(\frac{1}{j}\right)}{2^j} \approx -\frac{j}{2^j} \). Now consider the series \( \sum_{j=1}^\infty -\frac{j}{2^j} \).

Apply the Ratio Test or Root Test to the series \( \sum \frac{j}{2^j} \) to determine convergence. Since \( 2^j \) grows faster than any polynomial \( j \), the series converges absolutely, and thus the original series converges.

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Convergence of Infinite Series

An infinite series converges if the sequence of its partial sums approaches a finite limit. Understanding convergence is essential to determine whether the sum of infinitely many terms results in a finite value or diverges to infinity or oscillates.

Behavior and Approximation of Functions for Small Arguments

Analyzing the behavior of functions like cotangent near zero helps simplify terms in a series. For small x, cot(x) can be approximated using series expansions, which aids in understanding the general term's behavior and applying convergence tests effectively.

Convergence Tests for Series (Comparison and Ratio Tests)

Tests such as the Ratio Test and Comparison Test help determine series convergence by comparing terms to known convergent series or analyzing the limit of term ratios. These tests are crucial for series involving complex terms like cot(–1/j)/2^j.

11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.∑ (from j = 1 to ∞)cot(–1 / j) / 2ʲ