Evaluate the integral. ∫[(x−(x^3 /2)+(x^5 /(2.4))−(x^7 /(2.4.6))+...)(1−(x^2 /2^2 )+(x^4 /(2^2 .4^2 ))−(x^6 /(2^2 .4^2 .6^2 ))+....)]dx for 0<x<∞ Please help


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Question Number 5835 by sanusihammed last updated on 31/May/16

$E v a l u a t e t h e i n t e g r a l .$ $\int [(x - \frac{x^{3}}{2} + \frac{x^{5}}{2.4} - \frac{x^{7}}{2.4.6} + . . .) (1 - \frac{x^{2}}{2^{2}} + \frac{x^{4}}{2^{2} . 4^{2}} - \frac{x^{6}}{2^{2} . 4^{2} . 6^{2}} + . . . .)] d x$ $f o r 0 < x < \infty$ $P l e a s e h e l p$
Commented byYozzii last updated on 31/May/16
$(1/((2n−2)!!))=(1/(2×4×6×8×10×...×(2n−2))) =(1/(2^(n−1) (1×2×3×4×5×...×(n−1)))) ∴(1/((2n)!!))=(1/(2^n n!)) ( n≥0) (2n)!!=2n{(2n−2)!!} Σ_(n=0) ^∞ (((−1)^n x^(2n+1) )/(2^n (n!)))=(√2)Σ_(n=0) ^∞ (((−1)^n x^(2n+1) )/(((√2))^(2n+1) n!)) =xΣ_(n=0) ^∞ (((−1)^n )/(n!))((x^2 /2))^n =xΣ_(n=0) ^∞ (1/(n!))(−(x^2 /2))^n Σ_(n=0) ^∞ (((−1)^n x^(2n+1) )/(2^n (n!)))=xexp(((−x^2 )/2))=−(d/dx)(e^(−x^2 /2) ) −−−−−−−−−−−−−−−−−−−−−−− 1−(x^2 /2^2 )+(x^4 /(2^2 4^2 ))−(x^6 /(2^2 4^2 6^2 ))+...=Σ_(n=0) ^∞ (((−1)^n x^(2n) )/((2^n n!)^2 )) =Σ_(n=0) ^∞ (1/((n!)^2 ))(−(x^2 /4))^n =Σ_(n=0) ^∞ {(1/(n!))(((−x)/2))^n }{(1/(n!))((x/2))^n } −−−−−−−−−−−−−−−−−−−−−− ∴J=∫{xexp(−0.5x^2 )(Σ_(n=0) ^∞ (((−1)^n x^(2n) )/((2^n n!)^2 )))}dx J=Σ_(n=0) ^∞ [(((−1)^n )/((2^n n!)^2 )){∫(xe^(−0.5x^2 ) )x^(2n) dx}] =Σ_(n=0) ^∞ (((−1)^n )/((2^n n!)^2 )){−e^(−0.5x^2 ) x^(2n) +2n∫xe^(−0.5x^2 ) x^(2n−2) dx} =Σ_(n=0) ^∞ (((−1)^n )/((2^n n!)^2 ))(−x^(2n) e^(−0.5x^2 ) −2nx^(2n−2) e^(−0.5x^2 ) −2n(2n−2)x^(2n−4) e^(−0.5x^2 ) +2n(2n−2)(2n−4)∫x^(2n−6) xe^(−0.5x^2 ) dx) =Σ_(n=0) ^∞ (((−1)^n )/(((2n)!!)^2 ))(−e^(−0.5x^2 ) (x^(2n) +2nx^(2(n−1)) +2n(2n−2)x^(2(n−2)) +2n(2n−2)(2n−4)x^(2(n−3)) +....+(2n)!!x^(2(n−n)) ))+C J=Σ_(n=0) ^∞ ((((−1)^(n+1) e^(−0.5x^2 ) )/(((2n)!!)^2 )){Σ_(k=0) ^n x^(2(n−k)) (((2n)!!)/((2(n−k))!!))})+C J=e^(−0.5x^2 ) Σ_(n=0) ^∞ ((((−1)^(n+1) )/((2n)!!)){Σ_(k=0) ^n (1/((2(n−k))!!))x^(2(n−k)) })+C J=−e^(−0.5x^2 ) Σ_(n=0) ^∞ ((1/(n!))(((−x^2 )/2))^n {Σ_(k=0) ^n (1/((n−k)!))((x^2 /2))^(n−k) }) J=−e^(−0.5x^2 ) Σ_(n=0) ^∞ Σ_(k=0) ^n {(1/(n!(n−k)!))(((−x^2 )/2))^n ((x^2 /2))^(n−k) } J=−e^(−0.5x^2 ) Σ_(n=0) ^∞ ((1/(n!))(((−x^2 )/2))^n )(Σ_(k=0) ^∞ (1/((n−k)!))((x^2 /2))^(n−k) )+C ??? J=−e^(−0.5x^2 ) e^(−0.5x^2 ) e^(0.5x^2 ) +C J=−e^(−0.5x^2 ) +C J∣_0 ^∞ =−e^(−0.5x^2 ) ∣_0 ^∞ =1.$
$\frac{1}{(2 n - 2)!!} = \frac{1}{2 \times 4 \times 6 \times 8 \times 10 \times . . . \times (2 n - 2)}$ $= \frac{1}{2^{n - 1} (1 \times 2 \times 3 \times 4 \times 5 \times . . . \times (n - 1))}$ $∴ \frac{1}{(2 n)!!} = \frac{1}{2^{n} n!} (n ⩾ 0)$ $(2 n)!! = 2 n {(2 n - 2)!!}$ $\underset{n = 0}{\sum^{\infty}} \frac{{(- 1)}^{n} x^{2 n + 1}}{2^{n} (n!)} = \sqrt{2} \underset{n = 0}{\sum^{\infty}} \frac{{(- 1)}^{n} x^{2 n + 1}}{{(\sqrt{2})}^{2 n + 1} n!}$ $= x \underset{n = 0}{\sum^{\infty}} \frac{{(- 1)}^{n}}{n!} {(\frac{x^{2}}{2})}^{n}$ $= x \underset{n = 0}{\sum^{\infty}} \frac{1}{n!} {(- \frac{x^{2}}{2})}^{n}$ $\underset{n = 0}{\sum^{\infty}} \frac{{(- 1)}^{n} x^{2 n + 1}}{2^{n} (n!)} = x e x p (\frac{- x^{2}}{2}) = - \frac{d}{d x} (e^{- x^{2} / 2})$ $- - - - - - - - - - - - - - - - - - - - - - -$ $1 - \frac{x^{2}}{2^{2}} + \frac{x^{4}}{2^{2} 4^{2}} - \frac{x^{6}}{2^{2} 4^{2} 6^{2}} + . . . = \underset{n = 0}{\sum^{\infty}} \frac{{(- 1)}^{n} x^{2 n}}{{(2^{n} n!)}^{2}}$ $= \underset{n = 0}{\sum^{\infty}} \frac{1}{{(n!)}^{2}} {(- \frac{x^{2}}{4})}^{n} = \underset{n = 0}{\sum^{\infty}} {\frac{1}{n!} {(\frac{- x}{2})}^{n}} {\frac{1}{n!} {(\frac{x}{2})}^{n}}$ $- - - - - - - - - - - - - - - - - - - - - -$ $∴ J = \int {x e x p (- 0.5 x^{2}) (\underset{n = 0}{\sum^{\infty}} \frac{{(- 1)}^{n} x^{2 n}}{{(2^{n} n!)}^{2}})} d x$ $J = \underset{n = 0}{\sum^{\infty}} [\frac{{(- 1)}^{n}}{{(2^{n} n!)}^{2}} {\int ({x e}^{- 0.5 x^{2}}) x^{2 n} d x}]$ $= \underset{n = 0}{\sum^{\infty}} \frac{{(- 1)}^{n}}{{(2^{n} n!)}^{2}} {- e^{- 0.5 x^{2}} x^{2 n} + 2 n \int {x e}^{- 0.5 x^{2}} x^{2 n - 2} d x}$ $= \underset{n = 0}{\sum^{\infty}} \frac{{(- 1)}^{n}}{{(2^{n} n!)}^{2}} (- x^{2 n} e^{- 0.5 x^{2}} - 2 {n x}^{2 n - 2} e^{- 0.5 x^{2}} - 2 n (2 n - 2) x^{2 n - 4} e^{- 0.5 x^{2}} + 2 n (2 n - 2) (2 n - 4) \int x^{2 n - 6} {x e}^{- 0.5 x^{2}} d x)$ $= \underset{n = 0}{\sum^{\infty}} \frac{{(- 1)}^{n}}{{((2 n)!!)}^{2}} (- e^{- 0.5 x^{2}} (x^{2 n} + 2 {n x}^{2 (n - 1)} + 2 n (2 n - 2) x^{2 (n - 2)} + 2 n (2 n - 2) (2 n - 4) x^{2 (n - 3)} + . . . . + (2 n)!! x^{2 (n - n)})) + C$ $J = \underset{n = 0}{\sum^{\infty}} (\frac{{(- 1)}^{n + 1} e^{- 0.5 x^{2}}}{{((2 n)!!)}^{2}} {\underset{k = 0}{\sum^{n}} x^{2 (n - k)} \frac{(2 n)!!}{(2 (n - k))!!}}) + C$ $J = e^{- 0.5 x^{2}} \underset{n = 0}{\sum^{\infty}} (\frac{{(- 1)}^{n + 1}}{(2 n)!!} {\underset{k = 0}{\sum^{n}} \frac{1}{(2 (n - k))!!} x^{2 (n - k)}}) + C$ $J = - e^{- 0.5 x^{2}} \underset{n = 0}{\sum^{\infty}} (\frac{1}{n!} {(\frac{- x^{2}}{2})}^{n} {\underset{k = 0}{\sum^{n}} \frac{1}{(n - k)!} {(\frac{x^{2}}{2})}^{n - k}})$ $J = - e^{- 0.5 x^{2}} \underset{n = 0}{\sum^{\infty}} \underset{k = 0}{\sum^{n}} {\frac{1}{n! (n - k)!} {(\frac{- x^{2}}{2})}^{n} {(\frac{x^{2}}{2})}^{n - k}}$ $J = - e^{- 0.5 x^{2}} \underset{n = 0}{\sum^{\infty}} (\frac{1}{n!} {(\frac{- x^{2}}{2})}^{n}) (\underset{k = 0}{\sum^{\infty}} \frac{1}{(n - k)!} {(\frac{x^{2}}{2})}^{n - k}) + C ? ? ?$ $J = - e^{- 0.5 x^{2}} e^{- 0.5 x^{2}} e^{0.5 x^{2}} + C$ $J = - e^{- 0.5 x^{2}} + C$ $J ∣_{0}^{\infty} = - e^{- 0.5 x^{2}} ∣_{0}^{\infty} = 1 .$
Commented bysanusihammed last updated on 31/May/16

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