TS EAMCET · Maths · Definite Integration
\(\lim _{n \rightarrow \infty} \frac{1}{n^2}\left[e^{1 / n}+2 e^{2 / n}+3 e^{3 / n}+\ldots. .+2 n e^2\right]=\)
- A \(e^2-1\)
- B \(e^2+1\)
- C \(2 e^2-2\)
- D \(2 e^2+1\)
Answer & Solution
Correct Answer
(B) \(e^2+1\)
Step-by-step Solution
Detailed explanation
\( \lim _{n \rightarrow \infty} \frac{1}{n^2} \sum_{k=1}^{2n} k e^{k/n} = \lim _{n \rightarrow \infty} \frac{1}{n} \sum_{k=1}^{2n} \left(\frac{k}{n}\right) e^{k/n} \) \( = \int_0^2 x e^x dx \) \( = \left[x e^x - e^x\right]_0^2 \) \( = (2 e^2 - e^2) - (0 \cdot e^0 - e^0) \)…
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