55–70. More sequences
Find the limit of the following sequences or determine that the sequence diverges.

{(75n⁻¹ / 99ⁿ) + (5ⁿsinn / 8ⁿ)}

검증된 단계별 안내

Identify the given sequence as \( a_n = \frac{75}{n \cdot 99^n} + \frac{5^n \sin n}{8^n} \). We want to find \( \lim_{n \to \infty} a_n \) or determine if it diverges.

Analyze the first term \( \frac{75}{n \cdot 99^n} \): since \( 99^n \) grows exponentially and \( n \) grows linearly, this term approaches zero as \( n \to \infty \).

Analyze the second term \( \frac{5^n \sin n}{8^n} \): rewrite it as \( \sin n \cdot \left( \frac{5}{8} \right)^n \). Since \( \left( \frac{5}{8} \right)^n \) is an exponential decay (because \( \frac{5}{8} < 1 \)) and \( \sin n \) is bounded between -1 and 1, this term also approaches zero as \( n \to \infty \).

Combine the limits of both terms: since both approach zero, the sum \( a_n \) approaches zero as \( n \to \infty \).

Conclude that the sequence converges and its limit is zero.

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영상 길이:

주요 개념

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

Limits of Sequences

The limit of a sequence describes the value that the terms approach as the index n goes to infinity. Understanding how to evaluate limits helps determine whether a sequence converges to a finite number or diverges. Techniques often involve analyzing the behavior of individual terms or applying limit laws.

Exponential Growth and Decay

Exponential terms like 99ⁿ, 5ⁿ, and 8ⁿ grow or decay rapidly depending on the base. Comparing bases helps determine which terms dominate as n increases. For example, if the denominator grows faster than the numerator, the fraction tends to zero, influencing the sequence's limit.

Behavior of Oscillatory Functions in Sequences

Functions like sin(n) oscillate between -1 and 1, causing terms to fluctuate. When multiplied by terms that tend to zero, the oscillations become negligible, often leading to convergence. Understanding how oscillatory factors interact with exponential terms is key to analyzing sequence limits.

55–70. More sequencesFind the limit of the following sequences or determine that the sequence diverges.{(75n⁻¹ / 99ⁿ) + (5ⁿsinn / 8ⁿ)}