Factorials are fun

Calculus Level 2

What is the value of lim ⁡ n → ∞ ( n ! ) n n ( n ! ) \displaystyle \lim_{n\to \infty } \frac{ (n!)^n }{ n^{(n!)}} ?


The answer is 0.

Sachin Vishwakarma by Sachin Vishwakarma , 22, India

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2 solutions

Dallin Richards
Mar 13, 2016

The denominator, having the larger exponent, will grow significantly faster than the numerator until the numerator becomes negligible, thus making the limit equal zero.

2 Helpful 0 Interesting 0 Brilliant 0 Confused

Applying Stirling's formula and exponents rule n ! ∼ 2 π n ( n e ) n n! \sim \sqrt{2 \pi n}\left(\frac{n}{e}\right)^n

2 Helpful 0 Interesting 0 Brilliant 0 Confused

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