proof-that-n-gt-n-3-n-n-N-

Question Number 658 by 123456 last updated on 22/Feb/15

p r o o f t h a t n! > {(\frac{n}{3})}^{n}, n \in N^{*}

Commented by 123456 last updated on 20/Feb/15

n = 1 \Rightarrow 1! = 1 > \frac{1}{3} = {(\frac{1}{3})}^{1}

n = 1 \Rightarrow 0! = 1 > \frac{1}{3} \approx 0.33

n = 2 \Rightarrow 2! = 2 > \frac{4}{9} = {(\frac{2}{3})}^{2}

n = 2 \Rightarrow 1! = 1 > \frac{4}{9} \approx 0.44

n = 3 \Rightarrow 3! = 6 > 1 = {(\frac{3}{3})}^{3}

n = 3 \Rightarrow 2! = 2 > 1

n = 4 \Rightarrow 4! = 24 > \frac{256}{81} = {(\frac{4}{3})}^{4}

n = 4 \Rightarrow 3! = 6 > \frac{256}{81} \approx 3.16

n = 5 \Rightarrow 5! = 120 > \frac{3125}{243} = {(\frac{5}{3})}^{5}

n = 5 \Rightarrow 4! = 24 > \frac{3125}{243} \approx 13.23

n = 6 \Rightarrow 6! = 720 > 64 = {(\frac{6}{3})}^{6}

n = 6 \Rightarrow 5! = 120 > 64

n = 7 \Rightarrow 7! = 5040 > \frac{823543}{2187} = {(\frac{7}{3})}^{7}

n = 7 \Rightarrow 6! = 720 > \frac{823543}{2187} \approx 376.52

Commented by 123456 last updated on 20/Feb/15

k! > \frac{k^{k}}{3^{k}} \Rightarrow (k + 1)! > \frac{k^{k} (k + 1)}{3^{k}}

k! > \frac{k^{k}}{3^{k}} \Rightarrow \frac{k! {(k + 1)}^{k + 1}}{3 k^{k}} > \frac{{(k + 1)}^{k + 1}}{3^{k + 1}}

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