Limits
Limit n→∞ of n1[(n+1)(n+2)...(n+n)]1/n is:
Select the correct option:
Solution
4/e
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Let L be the limit. Simplify term: L=limn→∞[n(n+1)n(n+2)...n(n+n)]1/n L=lim∏r=1n(1+r/n)1/n.
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Take log: lnL=limn1∑r=1nln(1+r/n).
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Convert to Integral: lnL=∫01ln(1+x)dx.
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Evaluate Integral: ∫ln(1+x)dx=[(1+x)ln(1+x)−x]01 =(2ln2−1)−(0−0)=ln4−1=ln4−lne=ln(4/e).
So L=4/e.
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About This Question
- Subject
- mathematics
- Chapter
- limit, continuity and differentiability
- Topic
- limits
- Difficulty
- Medium
- Year
- 2025
Solution
Correct Answer:
4/e
-
Let L be the limit. Simplify term: L=limn→∞[n(n+1)n(n+2)...n(n+n)]1/n L=lim∏r=1n(1+r/n)1/n.
-
Take log: lnL=limn1∑r=1nln(1+r/n).
-
Convert to Integral: lnL=∫01ln(1+x)dx.
-
Evaluate Integral: ∫ln(1+x)dx=[(1+x)ln(1+x)−x]01 =(2ln2−1)−(0−0)=ln4−1=ln4−lne=ln(4/e).
So L=4/e.
This medium difficulty mathematics question is from the chapter limit, continuity and differentiability, covering the topic of limits. It appeared in the 2025 exam.
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