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I want to show the following equalities for harmonic sum $$\sum_{k=1}^n\frac{1}{k}=\log n-\int_{1}^n\frac{x-[x]}{x^2}dx+1$$

$$\sum_{k=1}^{\infty}\frac{(-1)^k}{k^3}=\lim_{n\to\infty}\int_0^{2n}\frac{-3}{x^4}([x]-2[\frac{x}{2}])dx$$

Any idea?

I want to show the following equalities for harmonic sum

$$\sum_{k=1}^{\infty}\frac{(-1)^k}{k^3}=\lim_{n\to\infty}\int_0^{2n}\frac{-3}{x^4}([x]-2[\frac{x}{2}])dx$$

Any idea?

I want to show thefollowing equality for harmonic sum $$\sum_{k=1}^n\frac{1}{k}=\log n-\int_{1}^n\frac{x-[x]}{x^2}dx+1$$

Any idea?

I want to show the following equalities for harmonic sum $$\sum_{k=1}^n\frac{1}{k}=\log n-\int_{1}^n\frac{x-[x]}{x^2}dx+1$$

$$\sum_{k=1}^{\infty}\frac{(-1)^k}{k^3}=\lim_{n\to\infty}\int_0^{2n}\frac{-3}{x^4}([x]-2[\frac{x}{2}])dx$$

Any idea?