Assume that bₙ is a sequence of positive numbers converging to 1/3. Determine if the following series converge or diverge.
a. ∑ (from n = 1 to ∞) [(bₙ₊₁ + bₙ) / n 4ⁿ]

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Identify the general term of the series: \( a_n = \frac{b_{n+1} + b_n}{n 4^n} \). Since \( b_n \) converges to \( \frac{1}{3} \), both \( b_n \) and \( b_{n+1} \) approach \( \frac{1}{3} \) as \( n \to \infty \).

Use the limit of \( b_n \) to approximate the behavior of the numerator for large \( n \): \( b_{n+1} + b_n \to \frac{1}{3} + \frac{1}{3} = \frac{2}{3} \). This helps simplify the term for large \( n \).

Focus on the denominator \( n 4^n \), which grows very rapidly due to the exponential term \( 4^n \). This suggests the terms \( a_n \) decrease quickly.

Apply the Comparison Test or Limit Comparison Test by comparing \( a_n \) to a known convergent series, such as \( \frac{1}{4^n} \), since \( \frac{b_{n+1} + b_n}{n 4^n} \) behaves similarly to \( \frac{constant}{n 4^n} \).

Conclude about convergence: since \( \sum \frac{1}{4^n} \) converges (geometric series with ratio \( \frac{1}{4} < 1 \)) and \( \frac{1}{n} \) grows slowly, the given series converges by comparison.

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주요 개념

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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 involves analyzing the behavior of terms as n approaches infinity, ensuring the sum does not diverge to infinity or oscillate indefinitely.

Limit of a Sequence

The limit of a sequence is the value that its terms approach as n becomes very large. Knowing that bₙ converges to 1/3 helps approximate terms in the series for large n, which is crucial for applying convergence tests effectively.

Comparison and Root/Ratio Tests for Series

These tests help determine series convergence by comparing terms to known convergent series or analyzing the nth root or ratio of successive terms. Given the presence of 4ⁿ in the denominator, the ratio test is particularly useful to assess the exponential decay of terms.

Assume that bₙ is a sequence of positive numbers converging to 1/3. Determine if the following series converge or diverge.a. ∑ (from n = 1 to ∞) [(bₙ₊₁ + bₙ) / n 4ⁿ]