Let [ ] denote the greatest integer function and f (x )= lim_n 0.05em 1n^3 _k=1^n 0.05em ( k^23^x ). Then 12 _j=1^ f (j ) is equal to \_ \_ \_ \_ .

Question

Accepted Answer

Correct answer: 2. (2) Let [ ⋅ denote the greatest integer function. We have f (x )= lim_n 0.05em 1n^3 _k=1^n 0.05em ( k^23^x ). For large n : _k=1^n 0.05em ( k^23^x ) ≈ _k=1^n 0.05em k^23^x= 13^x n (n+1 ) (2n+1 )6 ≈ n^33 3^x Therefore: f (x )= lim_n 0.05em 1n^3 n^33 3^x= 13^x+1 Computing the infinite series: _j=1^ f (j )= _j=1^ 0.05em 13^j+1= 13^2+ 13^3+ 13^4+ ⋯ = 19 (1+ 13+ 19+ ⋯)= 19 11-1/3= 19 32= 16Therefore: 12 _j=1^ f (j )=12 16=2

$Let [\cdot] denote the greatest integer function and f (x) = lim_{n \to \infty} \frac{1}{n ^{3}} \sum_{k = 1}^{n} (\frac{k ^{2}}{3 ^{x}}) . Then 12 \sum_{j = 1}^{\infty} f (j) is equal to____.$

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