DLMF:5.17.E7 (Q2187)

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DLMF:5.17.E7
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    Statements

    test
    C = lim n → ∞ ⁡ ( ∑ k = 1 n k ⁢ ln ⁡ k - ( 1 2 ⁢ n 2 + 1 2 ⁢ n + 1 12 ) ⁢ ln ⁡ n + 1 4 ⁢ n 2 ) = γ + ln ⁡ ( 2 ⁢ π ) 12 - ζ ′ ⁡ ( 2 ) 2 ⁢ π 2 = 1 12 - ζ ′ ⁡ ( - 1 ) , 𝐶 subscript → 𝑛 superscript subscript 𝑘 1 𝑛 𝑘 𝑘 1 2 superscript 𝑛 2 1 2 𝑛 1 12 𝑛 1 4 superscript 𝑛 2 2 𝜋 12 diffop Riemann-zeta 1 2 2 superscript 𝜋 2 1 12 diffop Riemann-zeta 1 1 {\displaystyle{\displaystyle C=\lim_{n\to\infty}\left(\sum_{k=1}^{n}k\ln k-% \left(\tfrac{1}{2}n^{2}+\tfrac{1}{2}n+\tfrac{1}{12}\right)\ln n+\tfrac{1}{4}n^% {2}\right)=\frac{\gamma+\ln\left(2\pi\right)}{12}-\frac{\zeta'\left(2\right)}{% 2\pi^{2}}=\frac{1}{12}-\zeta'\left(-1\right),}}
    0 references
    DLMF id
    DLMF:5.17.E7
    0 references
    Symbols used
    Euler’s constant
    test
    γ {\displaystyle{\displaystyle\gamma}}
    xml-id
    C5.S2.E3.m2aadec
    0 references
    Riemann zeta function
    test
    ζ ⁡ ( s ) Riemann-zeta 𝑠 {\displaystyle{\displaystyle\zeta\left(\NVar{s}\right)}}
    xml-id
    C25.S2.E1.m2adec
    0 references
     
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