TS EAMCET · Maths · Definite Integration
If \(f(n)=\frac{1}{n}[(n+1)(n+2)(n+3) \ldots(2 n)]^{\frac{1}{n}}\). then \(\lim _{n \rightarrow \infty} f(n)=\)
- A \(\frac{4}{e}\)
- B \(\log \left(\frac{4}{e}\right)\)
- C \(\frac{2}{e}\)
- D \(\log \left(\frac{2}{e}\right)\)
Answer & Solution
Correct Answer
(A) \(\frac{4}{e}\)
Step-by-step Solution
Detailed explanation
\begin{aligned} & \text { Let } A=\lim _{n \rightarrow \infty} \frac{1}{n}[(n+1)(n+2)(n+3) \ldots(2 n)]^{\frac{1}{n}} \\ & =\lim _{n \rightarrow \infty}\left[\left(1+\frac{1}{n}\right)\left(1+\frac{2}{n}\right) \ldots\left(1+\frac{n}{n}\right)\right]^{\frac{1}{n}} \\ &…
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