If `lim_(xtoI^(-)) prod_(n=0)^(oo)((1+x^(n+1))/(1+x^(n))^(x^n))=l` then `[1/l]` is equal to __________

Answer

1

Explanation

`(1+x^(n+1))/(1+x^(n))=1+(x^(n)(x-1))/(1+x^(n))`
So, `"In" (Pi_(n=0)^(N) (1+(x^(n)(x-1))/(1+x^(n)))^(x^(n)))=sum_(n=0)^(N) x^(n)"In"(1+(x^(n)(x-1))/(1+x^(n)))`
`le(x-1)(1/(1-x^(2))-1/(1-x^(3))+………-1/(1-x^(2k+1)))`
so, limit `=2/e`