AP EAMCET · Maths · Limits
\(\lim _{n \rightarrow \infty}\left(\frac{1^2}{n^3+1^3}+\frac{2^2}{n^3+2^3}+\ldots+\frac{1}{2 n}\right)=\)
- A \(\log 2\)
- B \(2 \log 2\)
- C \(\frac{1}{2} \log 2\)
- D \(\log \sqrt[3]{2}\)
Answer & Solution
Correct Answer
(D) \(\log \sqrt[3]{2}\)
Step-by-step Solution
Detailed explanation
\(L=\lim _{n \rightarrow \infty}\left[\frac{1^2}{n^3+1^3}+\frac{2^2}{n^3+2^3}+\ldots+\frac{n^2}{n^3+n^3}\right]\) \(=\lim _{n \rightarrow \infty} \sum_{r=1}^n \frac{r^2}{n^3+r^3}=\lim _{n \rightarrow \infty} \sum_{r=1}^n \frac{r^2}{n^3\left(1+\frac{r^3}{n^3}\right)}\)…
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