Value of (1+1/3)(1+1/(3^2))(1+1/(3^4))(1...

Value of `(1+1/3)(1+1/(3^2))(1+1/(3^4))(1+1/(3^8))oo` is equal to `3` b. `6/5` c. `3/2` d. none of these

A

3

B

`6/5`

C

`3/2`

D

none of these

Text Solution

Verified by Experts

The correct Answer is:

C

Consider the first product,
`P=(1+1/3)(1+1/3^(2))(1+1/3^(4))(1+1/3^(8))..(1+1/(3^(2^(n))))`
`=((1+1/3)(1+1/3^(2))(1+1/3^(4))(1+1/3^(8))...(1+1/3^(2^(n))))/((1-1/3))`
`=((1+1/3^(2))(1+1/3^(2))(1+1/3^(4))(1+1/3^(8))...(1+1/3^(2^(n))))/((1-1/3))`
`=((1+1/3^(4))(1+1/3^(4))(1+1/3^(8))...(1+1/3^(2^(n))))/((1-1/3))`
`1/((1-1/3))(1-(1/3)^(2^(n-1)))`
`=3/2(1-(1/3)^(2^(n+1)))`
`rArr(1+1/3)(1+1/3^(2))(1+1/3^(4))(1+1/3^(8))...` infinity
`=lim_(ntoo)3/2(1-(1/3)^(2^(n+1)))=3/2`

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