let V_n = ∫_0 ^(1+(1/n)) ((x+1)/(√(2x^2 +3))) dx 1) calculate lim_(n→+∞) V_n 2) find nature of the serie Σ V


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Question Number 57668 by maxmathsup by imad last updated on 09/Apr/19

$l e t V_{n} = \int_{0}^{1 + \frac{1}{n}} \frac{x + 1}{\sqrt{2 x^{2} + 3}} d x$ $1) c a l c u l a t e {l i m}_{n \to + \infty} V_{n}$ $2) f i n d n a t u r e o f t h e s e r i e Σ V_{n}$
Commented by maxmathsup by imad last updated on 10/Apr/19
$1) we have V_n =∫_0 ^(1+(1/n)) (x/(√(2x^2 +3)))dx +∫_0 ^(1+(1/n)) (dx/(√(2x^2 +3))) ∫_0 ^(1+(1/n)) (x/(√(2x^2 +3))) dx [(1/2)(√(2x^2 +3))]_0 ^(1+(1/n)) =(1/2){(√(2(1+(1/n))^2 +3))−(√3)} ∫_0 ^(1+(1/n)) (dx/(√(2x^2 +3))) =_((√2)x =(√3)u) ∫_0 ^(((√2)/(√3))(1+(1/n))) (1/((√3)(√(1+u^2 )))) ((√3)/(√2)) du =(1/(√2)) ∫_0 ^(((√2)/(√3))(1+(1/n))) (du/(√(1+u^2 ))) =[ln(u+(√(1+u^2 ))]_0 ^(((√2)/(√3))(1+(1/n))) =ln( (√(2/3))(1+(1/n))+(√(1+(2/3)(1+(1/n))^2 ))) ⇒ V_n =(1/2){ (√(2(1+(1/n))^2 +3))−(√3)}+ln{((√2)/(√3))(1+(1/n))+(√(1+(2/3)(1+(1/n))^2 ))} ⇒ lim_(n→+∞) V_n =(1/2){(√5)−(√3)} +ln{(((√2)+(√5))/(√3))} .$
$1) w e h a v e V_{n} = \int_{0}^{1 + \frac{1}{n}} \frac{x}{\sqrt{2 x^{2} + 3}} d x + \int_{0}^{1 + \frac{1}{n}} \frac{d x}{\sqrt{2 x^{2} + 3}}$ $\int_{0}^{1 + \frac{1}{n}} \frac{x}{\sqrt{2 x^{2} + 3}} d x {[\frac{1}{2} \sqrt{2 x^{2} + 3}]}_{0}^{1 + \frac{1}{n}} = \frac{1}{2} {\sqrt{2 {(1 + \frac{1}{n})}^{2} + 3} - \sqrt{3}}$ $\int_{0}^{1 + \frac{1}{n}} \frac{d x}{\sqrt{2 x^{2} + 3}} =_{\sqrt{2} x = \sqrt{3} u} \int_{0}^{\frac{\sqrt{2}}{\sqrt{3}} (1 + \frac{1}{n})} \frac{1}{\sqrt{3} \sqrt{1 + u^{2}}} \frac{\sqrt{3}}{\sqrt{2}} d u = \frac{1}{\sqrt{2}} \int_{0}^{\frac{\sqrt{2}}{\sqrt{3}} (1 + \frac{1}{n})} \frac{d u}{\sqrt{1 + u^{2}}}$ $= [l n {(u + \sqrt{1 + u^{2}}]}_{0}^{\frac{\sqrt{2}}{\sqrt{3}} (1 + \frac{1}{n})} = l n (\sqrt{\frac{2}{3}} (1 + \frac{1}{n}) + \sqrt{1 + \frac{2}{3} {(1 + \frac{1}{n})}^{2}}) \Rightarrow$ $V_{n} = \frac{1}{2} {\sqrt{2 {(1 + \frac{1}{n})}^{2} + 3} - \sqrt{3}} + l n {\frac{\sqrt{2}}{\sqrt{3}} (1 + \frac{1}{n}) + \sqrt{1 + \frac{2}{3} {(1 + \frac{1}{n})}^{2}}} \Rightarrow$ ${l i m}_{n \to + \infty} V_{n} = \frac{1}{2} {\sqrt{5} - \sqrt{3}} + l n {\frac{\sqrt{2} + \sqrt{5}}{\sqrt{3}}} .$
Commented by maxmathsup by imad last updated on 10/Apr/19

$2) Σ V_{n} d i v e r g e s b e c a u s e l i m V_{n} \neq 0 .$
Commented by Abdo msup. last updated on 10/Apr/19
$V_n =(1/2){(√(2(1+(1/n))^2 +3))−(√3)}+(1/(√2))ln{((√2)/(√3))(1+(1/n))+(√(1+(2/3)(1+(1/n))^2 ))} ⇒ lim_(n→+∞) V_n =(1/2){ (√5)−(√3)} +(1/(√2))ln((((√2) +(√5))/(√3))) .$
$V_{n} = \frac{1}{2} {\sqrt{2 {(1 + \frac{1}{n})}^{2} + 3} - \sqrt{3}} + \frac{1}{\sqrt{2}} l n {\frac{\sqrt{2}}{\sqrt{3}} (1 + \frac{1}{n}) + \sqrt{1 + \frac{2}{3} {(1 + \frac{1}{n})}^{2}}} \Rightarrow$ ${l i m}_{n \to + \infty} V_{n} = \frac{1}{2} {\sqrt{5} - \sqrt{3}} + \frac{1}{\sqrt{2}} l n (\frac{\sqrt{2} + \sqrt{5}}{\sqrt{3}}) .$
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