Answer by Paramanand Singh for Calculating pretty difficult limit that...
Extending the argument in this answer we can write $$\frac{1}{n}\sum_{k=1}^n f\left(\frac{k}{n}\right) =\int_0^1 f(x) \, dx+\frac{f(1)-f(0)}{2n}+\frac{f'(1)-f'(0)}{12n^2}+o(1/n^2)\tag{1}$$ Using $$f(x)...
View ArticleCalculating pretty difficult limit that invloves Riemann sums
Let $S_n = \sum_{k=1}^n\frac{1}{\sqrt{n^2+k^2}}$. Calculate the following limit$$\lim_{n \to \infty} n\left(n\Big(\ln(1+\sqrt{2})-S_n\Big)-\frac{1}{2\sqrt{2}\,(1+\sqrt{2})}\right).$$My intuition says...
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