Find the limits in Exercises 1–6.
5. lim(n→∞) (1/(n+1) + 1/(n+2) + ... + 1/(2n))

검증된 단계별 안내

Recognize that the expression is a sum of terms of the form $\frac{1}{k}$ where $k$ runs from $n+1$ to \$2n$. This can be written as $\sum_{k=n+1}^{2n} \frac{1}{k}$.

Recall that the harmonic series $H_m = \sum_{k=1}^m \frac{1}{k}$ and use it to rewrite the sum as $H_{2n} - H_n$.

Use the approximation for large $n$: $H_n \approx \ln(n) + \gamma$, where $\gamma$ is the Euler-Mascheroni constant, to express $H_{2n} - H_n$ as $\ln(2n) + \gamma - (\ln(n) + \gamma)$.

Simplify the expression by canceling out $\gamma$ and combining logarithms: $\ln(2n) - \ln(n) = \ln\left(\frac{2n}{n}\right) = \ln(2)$.

Conclude that the limit as $n \to \infty$ of the sum is $\ln(2)$, since the approximation becomes exact in the limit.

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.

Limit of a Sequence

The limit of a sequence describes the value that the terms of the sequence approach as the index goes to infinity. Understanding how to evaluate limits helps determine the behavior of sequences and series for very large indices.

Harmonic Series and Partial Sums

The harmonic series is the sum of reciprocals of natural numbers. Partial sums of the harmonic series, such as sums from 1 to n, grow without bound but slowly. Recognizing partial sums helps analyze expressions involving sums of terms like 1/k.

Integral Test and Approximation of Sums by Integrals

The integral test relates sums of sequences to definite integrals, allowing approximation of sums by integrals for large n. This technique is useful to estimate sums like 1/(n+1) + ... + 1/(2n) by comparing them to integrals of 1/x.

Find the limits in Exercises 1–6.5. lim(n→∞) (1/(n+1) + 1/(n+2) + ... + 1/(2n))