13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.

{bₙ}, where
bₙ = { n / (n + 1)if n ≤ 5000
ne⁻ⁿif n > 5000 }

검증된 단계별 안내

Understand the definition of the sequence \( b_n \): it has two different expressions depending on whether \( n \) is less than or equal to 5000 or greater than 5000.

Analyze the first part of the sequence for \( n \leq 5000 \): \( b_n = \frac{n}{n+1} \). Since this is a rational function, consider the limit as \( n \to \infty \) to understand its behavior.

Analyze the second part of the sequence for \( n > 5000 \): \( b_n = n e^{-n} \). Recognize that \( e^{-n} = \frac{1}{e^n} \) and consider the limit as \( n \to \infty \) to determine how this term behaves.

Since the sequence changes definition at \( n = 5000 \), check the limit of both parts as \( n \to \infty \) to see if they approach the same value or if the sequence diverges.

Conclude the overall limit of the sequence \( b_n \) by combining the results from both parts and considering the behavior for large \( n \).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Limit of a Sequence

The limit of a sequence is the value that the terms of the sequence approach as the index n goes to infinity. If the terms get arbitrarily close to a specific number, the sequence converges to that limit; otherwise, it diverges.

Piecewise-Defined Sequences

A piecewise-defined sequence has different formulas for different ranges of n. To find its limit, analyze the behavior of each piece separately, especially the part that applies as n approaches infinity.

Exponential Decay and Its Limit

Sequences involving terms like n·e⁻ⁿ combine polynomial growth with exponential decay. Since exponential decay dominates polynomial growth, such terms tend to zero as n approaches infinity.

13–52. Limits of sequencesFind the limit of the following sequences or determine that the sequence diverges.{bₙ}, where bₙ = { n / (n + 1)if n ≤ 5000 ne⁻ⁿif n > 5000 }