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The sum of the series (9)/(5^(2)*2*1)+(1...

The sum of the series `(9)/(5^(2)*2*1)+(13)/(5^(3)*3*2)+(17)/(5^(4)*4*3)+...` upto infinity

A

`((3)/(2))^((1)/(3))`

B

`((5)/(4))^((1)/(3))`

C

`((3)/(2))^((1)/(6))`

D

None of these

Text Solution

Verified by Experts

The correct Answer is:
A
`(a)` Let `S=1+(1)/(3^(2))+(1.4)/(2.4)(1)/(3^4)+(1.4.7)/(1.2.3)(1)/(3^(6))+.....`
`impliesS=1+((1)/(3))((1)/(3))+((1)/(3)*(1)/(3))/(1.2)((1)/(3))^(2)+((1)/(3)*(4)/(3)*(7)/(3))/(1.2.3)((1)/(3))^(2)+....`
`impliesS=1+((1)/(3))((1)/(3))+((1)/(3)(1+(1)/(3)))/(2!)((1)/(3))^(2)+((1)/(3)(1+(1)/(3))(2+(1)/(3)))/(3!)((1)/(3))^(3)+....`
`=(1-(1)/(3))^(-(1)/(3))=((2)/(3))^(-(1)/(3))=((3)/(2))^((1)/(3))`
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