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.

Expert Answer to "Evaluate lim n → ∞ e n 2..."

High School
Calculus and Analysis
Calculus 1
Limits and continuity

Shea Stuart

Answered question

2022-06-30

Evaluate

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

Answer & Explanation

Mateo Carson

Beginner2022-07-01Added 15 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.

Do you have a similar question?

Recalculate according to your conditions!

New Questions in Calculus 1

Find the local maximum and minimum values and saddle points of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function
$f (x, y) = x^{3} - 6 x y + 8 y^{3}$
$\frac{1}{\sec (x)}$ in derivative?
What is the derivative of $\ln (x + 1)$ ?
What is the limit of $e^{- x}$ as $x \to \infty$ ?
What is the derivative of $f (x) = 5^{\ln x}$ ?
What is the derivative of $e^{- 2 x}$ ?
How to find $lim \frac{e^{t} - 1}{t}$ as $t \to 0$ using l'Hospital's Rule?
What is the integral of $\sqrt{9 - x^{2}}$ ?
What is the derivative of $f (x) = \ln [x^{9} {(x + 3)}^{6} {(x^{2} + 7)}^{5}]$ ?
What Is the common difference or common ratio of the sequence 2, 5, 8, 11...?
How to find the derivative of $y = e^{5 x}$ ?
How to evaluate the limit $\frac{\sin (5 x)}{x}$ as x approaches 0?
How to find derivatives of parametric functions?
What is the antiderivative of $- 5 e^{x - 1}$ ?
How to evaluate: indefinite integral $\frac{1 + x}{1 + x^{2}} d x$ ?

Didn't find what you were looking for?