DLMF:2.10.E8 (Q995)

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DLMF:2.10.E8
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    Statements

    Defining formula
    ∑ j = 1 n - 1 1 j ∼ ln ⁡ n + γ - 1 2 ⁢ n - ∑ s = 1 ∞ B 2 ⁢ s 2 ⁢ s ⁢ 1 n 2 ⁢ s , asymptotic-expansion superscript subscript 𝑗 1 𝑛 1 1 𝑗 𝑛 1 2 𝑛 superscript subscript 𝑠 1 Bernoulli-number-B 2 𝑠 2 𝑠 1 superscript 𝑛 2 𝑠 {\displaystyle{\displaystyle\sum_{j=1}^{n-1}\frac{1}{j}\sim\ln n+\gamma-\frac{% 1}{2n}-\sum_{s=1}^{\infty}\frac{B_{2s}}{2s}\frac{1}{n^{2s}},}}
    0 references
    DLMF id
    DLMF:2.10.E8
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    constraint
    n → ∞ → 𝑛 {\displaystyle{\displaystyle n\to\infty}}
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    Symbols used
    Q11109
    Defining formula
    B n Bernoulli-number-B 𝑛 {\displaystyle{\displaystyle B_{\NVar{n}}}}
    xml-id
    C24.S2.SS1.m1aedec
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    Euler’s constant
    Defining formula
    γ {\displaystyle{\displaystyle\gamma}}
    xml-id
    C5.S2.E3.m2aadec
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    Q10931
    Defining formula
    ∼ asymptotic-expansion {\displaystyle{\displaystyle\sim}}
    xml-id
    C2.S1.SS3.p1.m11aadec
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    principal branch of logarithm function
    Defining formula
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2acdec
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    Q11113
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C2.S10.XMD3.m1fdec
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