Factorial Formula

Calculus Level 4

lim ⁡ n → ∞ ( n ! − 2 π n ( n e ) n ) = ? \large \lim_{n \to \infty} \left(n!- \sqrt{2{\pi}n}\left(\frac{n}{e}\right)^{n}\right)=?

2 π \sqrt{2{\pi}} 0 1 ∞ \infty

X X by X X , 17, Taiwan

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1 solution

X X
Jun 16, 2018

n ! = ( 2 π n ( n e ) n ) ( 1 + 1 12 n + 1 288 n 2 − 139 51840 n 3 − 571 2488320 n 4 + ⋯ ) n!=(\sqrt{2{\pi}n}(\frac{n}{e})^{n})(1+\frac1{12n}+\frac1{288n^2}-\frac{139}{51840n^3}-\frac{571}{2488320n^4}+\cdots) According to this,it is simple that the limit goes to ∞ \infty

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