Question

$\lim_{n \to \infty} \frac{(2n(2n-1)...(n+2)(n+1))^{1/n}}{n} =$

• A. $\int_0^1 \log x dx$
• B. $\int_0^1 x \log x dx$
• C. $\int_0^1 (x+1)\log(x+1) dx$
• D. $\int_0^1 \log(1+x) dx$

Hint:

Limits of products of the form $\lim_{n \to \infty} \left(\prod_{r=1}^n f(\frac{r}{n})\right)^{1/n}$ are a classic application of converting a limit to a definite integral. The standard procedure is to take the logarithm, which turns the product into a sum, and then recognize the sum as a Riemann sum for $\int_0^1 \ln(f(x)) dx$.

Answer 1

The Correct Option is D

Step 1: Rewrite the expression inside the limit.
Let $L$ be the given limit. The numerator is the product $(n+1)(n+2)\cdots(2n)$. \[ L = \lim_{n \to \infty} \frac{((n+1)(n+2)\cdots(2n))^{1/n}}{n}. \] Bring the denominator inside the $1/n$ power: \[ L = \lim_{n \to \infty} \left( \frac{(n+1)(n+2)\cdots(2n)}{n^n} \right)^{1/n}. \] Separate the product in the numerator: \[ L = \lim_{n \to \infty} \left( \left(\frac{n+1}{n}\right) \left(\frac{n+2}{n}\right) \cdots \left(\frac{n+n}{n}\right) \right)^{1/n}. \] This is $L = \lim_{n \to \infty} \left( \prod_{r=1}^{n} \left(1+\frac{r}{n}\right) \right)^{1/n}$.

Step 2: Take the natural logarithm to convert the product to a sum.
Let $y = \left( \prod_{r=1}^{n} \left(1+\frac{r}{n}\right) \right)^{1/n}$. \[ \ln y = \ln \left[ \left( \prod_{r=1}^{n} \left(1+\frac{r}{n}\right) \right)^{1/n} \right] = \frac{1}{n} \sum_{r=1}^{n} \ln\left(1+\frac{r}{n}\right). \]

Step 3: Recognize the limit of the sum as a definite integral.
The expression for $\ln y$ is a Riemann sum. The limit of a Riemann sum is a definite integral: \[ \lim_{n \to \infty} \frac{1}{n} \sum_{r=1}^{n} f\left(\frac{r}{n}\right) = \int_0^1 f(x)dx. \] In our case, the function is $f(x) = \ln(1+x)$. \[ \lim_{n \to \infty} \ln y = \int_0^1 \ln(1+x) dx. \] So, $\ln L = \int_0^1 \log(1+x) dx$.

Step 4: Relate the result to the given options.
The question asks for the value of the limit $L$. The options provided are definite integrals. This is a common format where the question implicitly asks for the integral representation of the limit's logarithm, or the question is simply asking which integral is related to the limit. Option (D) matches the expression we found for $\ln L$. Given the options, the question is not asking for the numerical value of the limit (which is $4/e$), but for the definite integral that arises in its calculation. Therefore, the intended answer is the integral itself.