Unfamiliar Limit
L = n → ∞ lim ( ln ( ( n 2 n 2 − 1 2 ) ⋅ ( n 2 n 2 − 2 2 ) ⋯ ( n 2 2 n − 1 ) ) 1 / n )
If L can be expressed in the form ln ( φ e α ) γ for φ , α , γ ∈ Z − { 1 } , then:
( φ + α + γ ) ! = ?
The answer is 6.
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1 solution
Technically, using Riemann sums is a bit problematic, since the integrand is unbounded. The proper Riemann integral of a bounded function on a bounded interval is defined through Riemann sums, and an improper Riemann integral such as this is defined by finding the limit of the integral from 0 to a as a → 1 . This double-limit construction does not fit well with this limit.
You can justify this identification as an integral, but you need Lebesgue integration and the Dominated Convergence Theorem to do so.
Avoiding Lebesgue integration, we can just spot that the quantity whose limit we want is ( 2 n 2 n − 1 ( 2 n ) ! ) n 1 and use Stirling's approximation.
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I agree with you , sir , by the way, i did by your method only,(stirling's app.) .
Exactly Same way I did!!! :-)
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Great... :-)
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from which books you get these amazing questions??? these are really cool like ur name Rishabh “cool" :p ;-)
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@Atul Shivam – No this one and most of the questions I post are original or inspirations which I mentioned at appropriate places.... I don't write 'This problem is original' because Brilliant is such a wide place where there is always a possibility of this and many other questions been posted before which I don't know... :-)
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@Rishabh Jain – Try also to post problems on limits that do not use Reimann sums as your calculus problems are becoming very much predictable (about the concept to be used to solve the problem).You could think of limits of recurring functions,squeeze theorem,and others so as to add variety to your problems.
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Gud... And yours??
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I am doing good in mgthematics and physics but bad in chemistry.
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@Rishabh Deep Singh – Organic Chemistry ? I'm suffering in physics!!
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@Rishabh Jain – I am good in organic chemistry but really bad in inorganic and physical chemistry.
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@Rishabh Deep Singh – Yep...I face a hard time memorising Inorganic chemistry.. :+(
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@Rishabh Jain – Are you on fb?
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@Rishabh Deep Singh – Yep... But no longer active.. :-(
@Rishabh Deep Singh – yeah solved it again , :-) @Rishabh Cool mention that you are giving jee in 2017 i guess ; while @Rishabh Deep Singh will be writing this year ??
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@Rudraksh Sisodia – No I'm giving it in 2016 only... :-P
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@Rishabh Jain – ohh really ,, then very best of luck ,, hope you rock the exam , and what about in organic ?
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@Rudraksh Sisodia – Ohh Thanks..... Ya I'm trying to memorise inorganic. :-)
Same solution again.
L = n → ∞ lim n 1 r = 0 ∑ n − 1 ( ln ( 1 − n 2 r 2 ) )
By Reimann Sums , this is:
∫ 0 1 ln ( 1 − x 2 ) d x
= ∫ 0 1 ln ( 1 − x ) d x + ∫ 0 1 ln ( 1 + x ) d x
= ∣ ( 1 − x ) ( ln ( 1 − x ) − 1 ) + ( 1 + x ) ( ln ( 1 + x ) − 1 ) ∣ 0 1
= ln ( 4 e − 2 ) = ln ( 2 e − 1 ) 2
∴ ( 2 + ( − 1 ) + 2 ) ! = 6