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Find the sum 1+2/6 + (2.5)/(6.12)+(2.5....

Find the sum ` 1+2/6 + (2.5)/(6.12)+(2.5.8)/(6.12.18)+.....oo`

Text Solution

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`1+2/6+(2*5)/(6*12)+(2*5*8)/(6*12*18)+....+oo`
`1+1/33*2/2+(2*5)/(2*3*3)*2^2/2^2+(2*5*8)/(6*12*18)*2^3/2^3+...+oo`
`1+(-1/2)(-2/3)+((-2/3)(-5/3))/(2!)*(-1/2)^2+...+oo`
`(1+x)^n=1+nx+(n(n+1))/(2!)x^2+...+oo`
`(1-x)^n=1-nx+(n(n+1))/(2!)*2`
`(1-1/2)^(-2/3)=1+(-1/2)(-2/3)+((-2/3)(-2/3-1))/(2!)(-1/2)^2`
`(1-1/2)^(-2/3)=(1/2)^(-2/3)`
`2^(2/3)`.
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