Evaluate lim n → ∞ e n 2 ( 2 n ) !

perhaps it becomes clearer that the limit is infinite, making the series expansion of the exponential e n 2 ( 2 n ) ! = ∑ m = 0 ∞ ( n 2 ) m m ! ( 2 n ) ! ≥ n 4 n ( ( 2 n ) ! ) 2 = ( n 2 n ( 2 n ) ! ) 2 the last expression on the right is ≥ C n C &gt; 0 for large n.

Question: "Evaluate lim n → ∞ e n 2..."

High School
Calculus and Analysis
Calculus 1
Limits and continuity

daktielti

Answered question

2022-06-29

Evaluate

lim_{n \to \infty} \frac{e^{n^{2}}}{(2 n)!}

Answer & Explanation

Alexzander Bowman

Beginner2022-06-30Added 19 answers

perhaps it becomes clearer that the limit is infinite, making the series expansion of the exponential

\frac{e^{n^{2}}}{(2 n)!} = \frac{\sum_{m = 0}^{\infty} \frac{(n^{2})^{m}}{m!}}{(2 n)!} \geq \frac{n^{4 n}}{((2 n)!)^{2}} = {(\frac{n^{2 n}}{(2 n)!})}^{2}

the last expression on the right is

\geq C n

C > 0

for large n.

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