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Evaluate lim(ntooo) n^(-n^(2))[(n+2^(0))...

Evaluate `lim_(ntooo) n^(-n^(2))[(n+2^(0))(n+2^(-1))(n+2^(-2))...(n+2^(-n+1))]^(n)`.

Text Solution

Verified by Experts

We have `underset(ntooo)limn^(-n^(2))[(n+1)(n+(1)/(2))...(n+(1)/(2^(n-1)))]^(n)`
`=underset(ntooo)lim[((n+1)(n+(1)/(2))...(n+(1)/(2^(n-1))))/(n^(n))]^(n)`
`=underset(ntooo)lim((n+1)/(n))^(n).((n+(1)/(2))/(n))^(n)...((n+(1)/(2^(n-1)))/(n))^(n)`
`=underset(ntooo)lim(1+(1)/(n))^(n).(1+(1)/(2n))^((2n)/(2))...(1+(1)/(2^(n-1)n))^((2^(n-1).n)/(2^(n-1)))`
`=underset(ntooo)lime^(1).e^((1)/(2)).e^(2^((1)/(2)))...e^((1)/(2^(n-1)))`
`=e^(1).e^((1)/(2)).e^(2^((1)/(2)))...("infinite factors")`
`=e^((1)/(1-(1)/(2)))=e^(2)`
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