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The value of the infinite product 6^(1/2...

The value of the infinite product `6^(1/2)xx6^(2/4)xx6^(3/8)xx6^(4/16)xx...`

A

6

B

36

C

216

D

`oo`

Text Solution

Verified by Experts

The correct Answer is:
B
`X=6^(1/2)xx6^(1/2)xx6^(3/8)xx6^(4/16)xx"........."oo.`
`=6^(((1)/(2)+(2)/(4)+(3)/(8)+(4)/(16)"............."oo).`
Let `S=(1)/(2)+(2)/(4)+(3)/(8)+(4)/(16)+"............."oo.`
`((1)/(2)S=+(1)/(4)+(2)/(8)+(3)/(16)+".........."oo.)/((1-(1)/(2))S=(1)/(2)+((2-1))/(4)+((3-2))/(8)+((4-3))/(16)+"............"oo.)`
`(S)/(2)=(1)/(2)+(1)/(4)+(1)/(8)+(1)/(16)+".............."oo.`
`(S)/(2)=((1)/(2))/(1-(1)/(2))=1.`
`S=2:.x=6^(2)=36.`
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