The discussion centers on the area summation problem under a curve, specifically analyzing the expression $$\frac{\sqrt{1}+\sqrt{2}+...+\sqrt{n}}{n^{3/2}}$$. The presence of ##n^{3/2}## in the denominator is justified by the need to normalize the sum of square roots as the number of terms increases. The correct formulation involves using Riemann sums, where $$S_n=\sum_k f(c_k) \Delta x$$, with $$c_k=k \cdot \Delta x$$ and $$f(c_k)=\sqrt{c_k}$$, leading to the conclusion that substituting $$\Delta x=\frac{1}{n}$$ is essential for accurate evaluation.
PREREQUISITESStudents and educators in mathematics, particularly those focusing on calculus and analysis, as well as anyone interested in understanding the principles of area under curves and summation techniques.
It should be ##\sum_k f(c_k) \Delta x##. Now ##c_k=k \cdot \Delta x\, , \,f(c_k)=\sqrt{c_k}## and ## \Delta x=\frac{1}{n}##. You simply stopped too soon before substituting ## \Delta x=\frac{1}{n}##.Karol said: