`1/(2.4)+ (1.3)/(2.4.6) + (1.3.5)/(2.4.6.8)+ (1.3.5.7)/(2.4.6.8.10)+....oo` is equal to

Explanation

as we know, `(1+x)^n = 1+ (n(n-1)x)/2 + (n(n-1)(n-2)x^2)/(3*2*1)+ ......`

`(1+x)^(-1/2) = 1 + ((-1/2)(-3/2)x)/2 + ((-1/2)(-3/2)(-5/2)x^2)/(3*2*1)+ .....`

`(1+x)^(-1/2) = 1 + (1*3*x)/(2*4) - (1*3*5x^2)/(2*4*6) + ...`

`1/2(1-x)^(-1/2) = 1 - x/2 + (1*3*x^2)/(2*4) + (1*3*5x^3)/(2*4*6) + ....`

`= 1/2 - x/2 + (1*3*x^2)/(2*4*2) + (1*3*5)/(2*4*6*2) x^3+ .....`

integrating both sides

`1/2 int_0^1 (1-x)^(-1/2) = 1/2 - [x^2/(2*4)] + (1*3)/(2*4*2)[x^2/3] + (1*3*5)/(2*4*6*2)[ x^4/4]`

`1/2 int_0^1 (1-x)^(-1/2) = 1/2 - [x^2/(2*4)]`

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