Q 158528 P=Π_(n=1) ^∞ ((((n+1)^3 −1)/((n+1)^3 +1))) ⇒ P = Π_(n=1) ^∞ ((((n+1)^3 −1^3 )/((n+1)^3 +1^3 ))) ⇒ P = Π_(n=1) ^∞ {(((n+1−1)(n^2 +2n+1+n+1+1))/((n+1+1)(n^2 +2n+1−n−1+1)))} ⇒ P = Π_(n=1) ^∞ {(n/(n+2))}•Π_(n=1) ^∞ {((n^2 +3n+3)/(n^2 +n+1))} = lim_(n→∞) Π_(k=1) ^n {(k/(k+2))}•lim_(n→∞) Π_(k=1) ^n {((k^2 +3k+3)/(k^2 +k+1))} = lim_(n→∞) {(1/3)•(2/4)•(3/5)•...•(n/(n+2))}×lim_(n→∞) {(7/3)•((13)/7)•...•((n^2 +3n+3)/(n^2 +n+1))} =2lim_(n→∞) {(1/((n+1)(n+2)))}×(1/3)lim_(n→∞) {n^2 +3n+3} =(2/3)lim_(n→∞) {((n^2 +3n+3)/(n^2 +3n+2))} = (2/3)lim_(n→∞) {((1+(3/n)+(3/n^2 ))/(1+(3/n)+(2/n^2 )))} = (2/3). P = Π_(n=1) ^∞ ((((n+1)^3 −1)/((n+1)^3 +1))) = (2/3).. ...............Le puissant...............


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Question Number 158751 by puissant last updated on 08/Nov/21
$Q 158528 P=Π_(n=1) ^∞ ((((n+1)^3 −1)/((n+1)^3 +1))) ⇒ P = Π_(n=1) ^∞ ((((n+1)^3 −1^3 )/((n+1)^3 +1^3 ))) ⇒ P = Π_(n=1) ^∞ {(((n+1−1)(n^2 +2n+1+n+1+1))/((n+1+1)(n^2 +2n+1−n−1+1)))} ⇒ P = Π_(n=1) ^∞ {(n/(n+2))}•Π_(n=1) ^∞ {((n^2 +3n+3)/(n^2 +n+1))} = lim_(n→∞) Π_(k=1) ^n {(k/(k+2))}•lim_(n→∞) Π_(k=1) ^n {((k^2 +3k+3)/(k^2 +k+1))} = lim_(n→∞) {(1/3)•(2/4)•(3/5)•...•(n/(n+2))}×lim_(n→∞) {(7/3)•((13)/7)•...•((n^2 +3n+3)/(n^2 +n+1))} =2lim_(n→∞) {(1/((n+1)(n+2)))}×(1/3)lim_(n→∞) {n^2 +3n+3} =(2/3)lim_(n→∞) {((n^2 +3n+3)/(n^2 +3n+2))} = (2/3)lim_(n→∞) {((1+(3/n)+(3/n^2 ))/(1+(3/n)+(2/n^2 )))} = (2/3). P = Π_(n=1) ^∞ ((((n+1)^3 −1)/((n+1)^3 +1))) = (2/3).. ...............Le puissant...............$
$Q 158528$ $P = \underset{n = 1}{\prod^{\infty}} (\frac{{(n + 1)}^{3} - 1}{{(n + 1)}^{3} + 1})$ $\Rightarrow P = \underset{n = 1}{\prod^{\infty}} (\frac{{(n + 1)}^{3} - 1^{3}}{{(n + 1)}^{3} + 1^{3}})$ $\Rightarrow P = \underset{n = 1}{\prod^{\infty}} {\frac{(n + 1 - 1) (n^{2} + 2 n + 1 + n + 1 + 1)}{(n + 1 + 1) (n^{2} + 2 n + 1 - n - 1 + 1)}}$ $\Rightarrow P = \underset{n = 1}{\prod^{\infty}} {\frac{n}{n + 2}} ∙ \underset{n = 1}{\prod^{\infty}} {\frac{n^{2} + 3 n + 3}{n^{2} + n + 1}}$ $= \lim_{n \to \infty} \underset{k = 1}{\prod^{n}} {\frac{k}{k + 2}} ∙ \lim_{n \to \infty} \underset{k = 1}{\prod^{n}} {\frac{k^{2} + 3 k + 3}{k^{2} + k + 1}}$ $= \lim_{n \to \infty} {\frac{1}{3} ∙ \frac{2}{4} ∙ \frac{3}{5} ∙ . . . ∙ \frac{n}{n + 2}} \times \lim_{n \to \infty} {\frac{7}{3} ∙ \frac{13}{7} ∙ . . . ∙ \frac{n^{2} + 3 n + 3}{n^{2} + n + 1}}$ $= 2 \lim_{n \to \infty} {\frac{1}{(n + 1) (n + 2)}} \times \frac{1}{3} \lim_{n \to \infty} {n^{2} + 3 n + 3}$ $= \frac{2}{3} \lim_{n \to \infty} {\frac{n^{2} + 3 n + 3}{n^{2} + 3 n + 2}} = \frac{2}{3} \lim_{n \to \infty} {\frac{1 + \frac{3}{n} + \frac{3}{n^{2}}}{1 + \frac{3}{n} + \frac{2}{n^{2}}}} = \frac{2}{3} .$ $P = \underset{n = 1}{\prod^{\infty}} (\frac{{(n + 1)}^{3} - 1}{{(n + 1)}^{3} + 1}) = \frac{2}{3} . .$ $. . . . . . . . . . . . . . . L e p u i s s a n t . . . . . . . . . . . . . . .$
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