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2.10.E7 - MaRDI portal

2.10.E7 (Q965)

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2.10.E7
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    Statements

    DLMF defining formula
    ∑ j = 1 n - 1 j α ∼ ζ ⁡ ( - α ) + n α + 1 α + 1 ⁢ ∑ s = 0 ∞ ( α + 1 s ) ⁢ B s n s , asymptotic-expansion superscript subscript 𝑗 1 𝑛 1 superscript 𝑗 𝛼 Riemann-zeta 𝛼 superscript 𝑛 𝛼 1 𝛼 1 superscript subscript 𝑠 0 binomial 𝛼 1 𝑠 Bernoulli-number-B 𝑠 superscript 𝑛 𝑠 {\displaystyle{\displaystyle\sum_{j=1}^{n-1}j^{\alpha}\sim\zeta\left(-\alpha% \right)+\frac{n^{\alpha+1}}{\alpha+1}\sum_{s=0}^{\infty}\genfrac{(}{)}{0.0pt}{% }{\alpha+1}{s}\frac{B_{s}}{n^{s}},}}
    0 references
    constraint
    n → ∞ → 𝑛 {\displaystyle{\displaystyle n\to\infty}}
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    Symbols used
    Q11083
    DLMF defining formula
    B n Bernoulli-number-B 𝑛 {\displaystyle{\displaystyle B_{\NVar{n}}}}
    xml-id
    C24.S2.SS1.m1addec
    0 references
    Riemann zeta function
    DLMF defining formula
    ζ ⁡ ( s ) Riemann-zeta 𝑠 {\displaystyle{\displaystyle\zeta\left(\NVar{s}\right)}}
    xml-id
    C25.S2.E1.m2aadec
    0 references
    Q10905
    DLMF defining formula
    ∼ asymptotic-expansion {\displaystyle{\displaystyle\sim}}
    xml-id
    C2.S1.SS3.p1.m11adec
    0 references
    Q10728
    DLMF defining formula
    ( m n ) binomial 𝑚 𝑛 {\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}}}
    xml-id
    C1.S2.SS1.m1adec
    0 references
    integer
    DLMF defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C2.S10.XMD3.m1edec
    0 references

    Identifiers

     
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