Determining Convergence or Divergence
Which of the series in Exercises 17–56 converge, and which diverge? Use any method, and give reasons for your answers.
∑ (from n=1 to ∞) (2ⁿ + 3ⁿ) / (3ⁿ + 4ⁿ)

검증된 단계별 안내

First, examine the general term of the series: $a_n = \frac{2^n + 3^n}{3^n + 4^n}$. To determine convergence or divergence, analyze the behavior of $a_n$ as $n$ approaches infinity.

Identify the dominant terms in the numerator and denominator for large $n$. Since exponential functions grow at different rates, compare \$3^n$ and $4^n$ in the denominator, and $2^n$ and $3^n$ in the numerator.

Simplify the term by dividing numerator and denominator by the largest exponential term in the denominator, which is \$4^n$. This gives: $a_n = \frac{\frac{2^n}{4^n} + \frac{3^n}{4^n}}{\frac{3^n}{4^n} + 1}$.

Rewrite the fractions inside the numerator and denominator using properties of exponents: $\left(\frac{2}{4}\right)^n$, $\left(\frac{3}{4}\right)^n$, and $\left(\frac{3}{4}\right)^n$. Analyze the limit of $a_n$ as $n \to \infty$ by considering these terms.

If $\lim_{n \to \infty} a_n \neq 0$, then by the Test for Divergence, the series diverges. If the limit is zero, consider applying other convergence tests such as the Comparison Test or Ratio Test to determine if the series converges.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

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

Infinite Series and Convergence

An infinite series is the sum of infinitely many terms. Determining whether a series converges means checking if the sum approaches a finite limit as the number of terms grows indefinitely. If the sum does not approach a finite value, the series diverges.

Limit Comparison Test

The Limit Comparison Test compares the terms of a given series with those of a known benchmark series. By taking the limit of the ratio of their terms, if the limit is a positive finite number, both series either converge or diverge together. This test is useful when terms are complex expressions.

Behavior of Exponential Terms in Series

When series terms involve exponential expressions like aⁿ, the dominant base (largest base in numerator and denominator) determines the term's behavior as n grows. Comparing dominant exponential terms helps simplify the series and decide convergence by analyzing the ratio of these dominant terms.

Determining Convergence or DivergenceWhich of the series in Exercises 17–56 converge, and which diverge? Use any method, and give reasons for your answers.∑ (from n=1 to ∞) (2ⁿ + 3ⁿ) / (3ⁿ + 4ⁿ)