Evaluate the limits in Exercise 9 and 10 by identifying them with definite integrals and evaluating the integrals.
lim (n → ∞) Σ (from k=1 to n) ln √(1 + k/n)

검증된 단계별 안내

Recognize that the given limit is a Riemann sum of the form \(\lim_{n \to \infty} \sum_{k=1}^n f\left(\frac{k}{n}\right) \Delta x\), where \(\Delta x = \frac{1}{n}\) and \(f(x) = \ln \sqrt{1 + x}\).

Rewrite the summand \(\ln \sqrt{1 + \frac{k}{n}}\) as \(\frac{1}{2} \ln \left(1 + \frac{k}{n}\right)\) to simplify the expression inside the sum.

Express the sum as \(\sum_{k=1}^n \frac{1}{2} \ln \left(1 + \frac{k}{n}\right) \cdot \frac{1}{n}\), which matches the form of a Riemann sum for the integral of \(\frac{1}{2} \ln(1 + x)\) over the interval \([0,1]\).

Set up the definite integral corresponding to the limit: \(\int_0^1 \frac{1}{2} \ln(1 + x) \, dx\).

Evaluate the integral \(\int_0^1 \frac{1}{2} \ln(1 + x) \, dx\) using integration techniques such as integration by parts to find the exact value representing the limit.

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주요 개념

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Definite Integral as a Limit of Riemann Sums

A definite integral can be interpreted as the limit of a sum of function values times small intervals, known as a Riemann sum. As the number of subintervals increases indefinitely, the sum approaches the exact area under the curve. Recognizing a limit of sums as a definite integral is key to evaluating such limits.

Properties of Logarithmic Functions

Logarithmic functions, such as ln(x), have properties that simplify expressions, like ln(√x) = (1/2)ln(x). Understanding these properties helps in rewriting the sum's terms into a more manageable form, facilitating the identification of the integrand in the limit process.

Limit of a Sequence and Continuity

Evaluating limits involving sequences often requires understanding how functions behave as their inputs approach a point. Continuity ensures that the limit of the function values equals the function value at the limit point, allowing substitution inside the integral after identifying the integrand.

Evaluate the limits in Exercise 9 and 10 by identifying them with definite integrals and evaluating the integrals.lim (n → ∞) Σ (from k=1 to n) ln √(1 + k/n)