let U_n = ∫_(1/n) ^(2/n) Γ(x)Γ(1−x)dx with n≥3 1) calculate and determine lim_(n→+∞) U_n 2) study the convergence of Σ U


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Question Number 65004 by mathmax by abdo last updated on 24/Jul/19

$l e t U_{n} = \int_{\frac{1}{n}}^{\frac{2}{n}} Γ (x) Γ (1 - x) d x w i t h n ⩾ 3$ $1) c a l c u l a t e a n d d e t e r m i n e {l i m}_{n \to + \infty} U_{n}$ $2) s t u d y t h e c o n v e r g e n c e o f Σ U_{n}$
Commented by mathmax by abdo last updated on 24/Jul/19

$1) w e h a v e Γ (x) . Γ (1 - x) = \frac{π}{s i n (π x)} \Rightarrow U_{n} = \int_{\frac{1}{n}}^{\frac{2}{n}} \frac{π}{s i n (π x)} d x$ $=_{π x = t} π \int_{\frac{π}{n}}^{\frac{2 π}{n}} \frac{d t}{π s i n t} = \int_{\frac{π}{n}}^{\frac{2 π}{n}} \frac{d t}{s i n t} =_{t a n (\frac{t}{2}) = u} \int_{t a n (\frac{π}{2 n})}^{t a n (\frac{π}{n})} \frac{1}{\frac{2 u}{1 + u^{2}}} \frac{2 d u}{1 + u^{2}}$ $= \int_{t a n (\frac{π}{2 n})}^{t a n (\frac{π}{n})} \frac{d u}{u} = {[l n ∣ u ∣]}_{t a n (\frac{π}{2 n})}^{t a n (\frac{π}{n})} = l n ∣ \frac{t a n (\frac{π}{n})}{t a n (\frac{π}{2 n})} ∣ \Rightarrow U_{n} = l n ∣ \frac{t a n (\frac{π}{n})}{t a n (\frac{π}{2 n})} ∣$ $w e h a v e t a n (\frac{π}{n}) \sim \frac{π}{n} a n d t a n (\frac{π}{2 n}) \sim \frac{π}{2 n} \Rightarrow$ $\frac{t a n (\frac{π}{n})}{t a n (\frac{π}{2 n})} \sim \frac{π}{n} \frac{2 n}{π} = 2 \Rightarrow {l i m}_{n \to + \infty} U_{n} = l n (2)$ $2) {l i m}_{n \to + \infty} U_{n} \neq 0 \Rightarrow Σ U_{n} d i v e r g e s .$
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