Q-158528-P-n-1-n-1-3-1-n-1-3-1-P-n-1-n-1-3-1-3-n-1-3-1-3-P-n-1-n-1-1-n-2-2n-1-n-1-1-n-1-1-n-2-2n-1-n-1-1-P-n-1-

Question Number 158751 by puissant last updated on 08/Nov/21

Q 158528

P = \underset{n = 1}{\prod^{\infty}} (\frac{{(n + 1)}^{3} - 1}{{(n + 1)}^{3} + 1})

\Rightarrow P = \underset{n = 1}{\prod^{\infty}} (\frac{{(n + 1)}^{3} - 1^{3}}{{(n + 1)}^{3} + 1^{3}})

\Rightarrow P = \underset{n = 1}{\prod^{\infty}} {\frac{(n + 1 - 1) (n^{2} + 2 n + 1 + n + 1 + 1)}{(n + 1 + 1) (n^{2} + 2 n + 1 - n - 1 + 1)}}

\Rightarrow P = \underset{n = 1}{\prod^{\infty}} {\frac{n}{n + 2}} ∙ \underset{n = 1}{\prod^{\infty}} {\frac{n^{2} + 3 n + 3}{n^{2} + n + 1}}

= \lim_{n \to \infty} \underset{k = 1}{\prod^{n}} {\frac{k}{k + 2}} ∙ \lim_{n \to \infty} \underset{k = 1}{\prod^{n}} {\frac{k^{2} + 3 k + 3}{k^{2} + k + 1}}

= \lim_{n \to \infty} {\frac{1}{3} ∙ \frac{2}{4} ∙ \frac{3}{5} ∙ \dots ∙ \frac{n}{n + 2}} \times \lim_{n \to \infty} {\frac{7}{3} ∙ \frac{13}{7} ∙ \dots ∙ \frac{n^{2} + 3 n + 3}{n^{2} + n + 1}}

= 2 \lim_{n \to \infty} {\frac{1}{(n + 1) (n + 2)}} \times \frac{1}{3} \lim_{n \to \infty} {n^{2} + 3 n + 3}

= \frac{2}{3} \lim_{n \to \infty} {\frac{n^{2} + 3 n + 3}{n^{2} + 3 n + 2}} = \frac{2}{3} \lim_{n \to \infty} {\frac{1 + \frac{3}{n} + \frac{3}{n^{2}}}{1 + \frac{3}{n} + \frac{2}{n^{2}}}} = \frac{2}{3} .

P = \underset{n = 1}{\prod^{\infty}} (\frac{{(n + 1)}^{3} - 1}{{(n + 1)}^{3} + 1}) = \frac{2}{3} . .

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