11it simplifies to
0∫1([2/x]-2[1/x])dx
after that [7]
Evaluate:
\lim_{n\to\infty}\dfrac{1}{n}\sum_{k=1}^n\left(\left[\dfrac{2n}{k}\right]-2\left[\dfrac{n}{k}\right]\right)
Here [x] represents the greatest integer less than or equal to .
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39 Answers
1I1 = \int_{0}^{1}{[\frac{2}{x}]dx}
=> I1 = \int_{2/3}^{1}{2.dx}+ \int_{1/2}^{2/3}{3.dx} +\int_{2/5}^{1/2}{4.dx}+ .....
=> I1 = 2(1- 2/3) + 3(2- 1/2) + 4(1/2 - 2/5) + .....
=> I1 = 2+ 2 + 2 +.... - (4/3 + 3/2 + 8/5 + 5/3 ....)
(cont.. in the nxt post..)