Question

Evaluate \( \lim_{n \to \infty} \left( \frac{1^2}{n^3 + 1^3} + \frac{2^2}{n^3 + 2^3} + \cdots + \frac{n^2}{n^3 + n^3} \right) \)

• A. \( \log 2 \)
• B. \( 2 \log 2 \)
• C. \( \frac{1}{2} \log 2 \)
• D. \( \log \sqrt{2} \)

Hint:

Converting Series to Riemann Sums}
Express sum as \( \sum \frac{1}{n} f\left( \frac{k}{n} \right) \)
Change sum to integral over [0,1]
Simplify definite integrals using substitution when necessary

Answer 1

The Correct Option is D

\[ \frac{k^2}{n^3 + k^3} = \frac{k^2}{n^3(1 + (k/n)^3)} \Rightarrow \sum_{k=1}^n \frac{k^2}{n^3 + k^3} \approx \sum_{k=1}^n \frac{1}{n} \cdot \frac{(k/n)^2}{1 + (k/n)^3} \] This is a Riemann sum: \[ \int_0^1 \frac{x^2}{1 + x^3} dx \Rightarrow \text{Substitute } x^3 = t \Rightarrow 3x^2 dx = dt \Rightarrow \frac{1}{3} \int_0^1 \frac{1}{1 + t} dt = \frac{1}{3} \log 2 \] Compare: \[ \text{Options } \Rightarrow \boxed{\log \sqrt{2} = \frac{1}{2} \log 2} \Rightarrow \text{Answer: } \boxed{\log \sqrt{2}} \]