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Another integral for $\pi$

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#calculus #integration #definite-integrals #pi #fractional-part
851
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Equality of sums with fractional parts of the form $\sum_{k=1}^{n}k\{\frac{mk}{n}\}$

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#number-theory #summation #fractional-part
268
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Evaluating an Indefinite Integral involving $\{x\} = x - \lfloor x \rfloor$

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#integration #indefinite-integrals #ceiling-and-floor-functions #fractional-part
406
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Evaluating $\sum_{x=a}^b \left\lfloor {\frac{k}{x}} \right\rfloor$

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#summation #ceiling-and-floor-functions #fractional-part
207
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Calculation of value of $ \int_{0}^{n}\cos\left(\lfloor x \rfloor\cdot \{x\}\right)dx\;,$

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#calculus #integration #ceiling-and-floor-functions #fractional-part
100
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Is the derivative of $\{x\}$ on $(0,1)$ always equal to $1$?

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#solution-verification #fractional-part
660
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A fractional part integral giving $\frac{F_{n-1}}{F_n}-\frac{(-1)^n}{F_n^2}\ln\left(\!\frac{F_{n+2}-F_n\gamma}{F_{n+1}-F_n\gamma}\right)$

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#calculus #integration #definite-integrals #euler-mascheroni-constant #fractional-part
2k
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Expressing n mod m in terms of floor values?

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#discrete-mathematics #ceiling-and-floor-functions #fractional-part
327
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fractional part of the square of natural number

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#calculus #sequences-and-series #fractional-part
139
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Bound for sum of fractional parts of $\sqrt{k}\, \alpha$

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#number-theory #reference-request #summation #asymptotics #fractional-part
307
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Rewriting the Floor function in terms of the Heaviside Step Function and/or Sign Function

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#functions #approximation #ceiling-and-floor-functions #fractional-part
33
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How to find if this function is periodic

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#functions #periodic-functions #fractional-part
46
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Continous, Non-Fourier, Trigonometric Approximation of the Fractional Part Function

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#trigonometry #approximation #fractional-part
233
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Closed expression for sum $\sum_{k = 1}^{\infty} \frac{\left\lfloor \sqrt{k} \right \rfloor}{k^2}$

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#sequences-and-series #harmonic-numbers #fractional-part
110
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Find $\lim\limits_{n \to \infty} n \int\limits_0^\pi \{ -x\}^n dx$.

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#calculus #integration #limits #definite-integrals #fractional-part

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