lim-n-S-i-1-n-i-n-1-L-S-R-S-gt-0-What-can-we-tell-about-L-Does-there-exist-a-limit-Is-it-positive-negative-

Question Number 5207 by FilupSmith last updated on 30/Apr/16

\lim_{n \to \infty} \frac{S - (\underset{i = 1}{\sum^{n}} i!)}{n} - 1 = L

S \in R, S > 0

What can we tell about L ?

Does there exist a limit ?

Is it positive / negative ?

Commented by FilupSmith last updated on 30/Apr/16

f = \underset{i = 1}{\sum^{n}} i! \Rightarrow \frac{d f}{d n} = \frac{d}{d n} (\underset{i = 1}{\sum^{n}} i!)

\lim_{n \to \infty} f = \infty

∴ \lim_{n \to \infty} \frac{S - (\underset{i = 1}{\sum^{n}} i!)}{n} - 1 \Rightarrow \frac{S - \infty}{\infty} - 1

∴ L + 1 = \lim_{n \to \infty} \frac{\frac{d}{d n} (S - (\underset{i = 1}{\sum^{n}} i!))}{\frac{d}{d n} n}

∴ L + 1 = \lim_{n \to \infty} \frac{1 - \frac{d f}{d n}}{1}

L = \lim_{n \to \infty} - \frac{d f}{d n}

What is \frac{d}{d n} (\underset{i = 1}{\sum^{n}} i!) ? ? ?

Commented by FilupSmith last updated on 30/Apr/16

\underset{i = 1}{\sum^{n}} i! = n! + (n - 1)! + (n - 2)! + \dots + 1!

Does :

\frac{d}{d n} (\underset{i = 1}{\sum^{n}} i!) = Γ (n + 1) ψ^{(0)} (n + 1) + Γ (n) ψ^{(0)} (n) + Γ (n - 1) ψ^{(0)} (n - 1) + \dots + Γ (2) ψ^{(0)} (2)

∴ \frac{d}{d n} (\underset{i = 1}{\sum^{n}} i!) = \underset{i = 1}{\sum^{n}} (Γ (i + 1) ψ^{(0)} (i + 1))