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