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Fungrim entry: 4aab8a

∑k=1mkn=Bn+1 ⁣(m+1)−Bm+1m+1\sum_{k=1}^{m} {k}^{n} = \frac{B_{n + 1}\!\left(m + 1\right) - B_{m + 1}}{m + 1}
Assumptions:n∈Z≥0  and⁡  m∈Z≥0n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; m \in \mathbb{Z}_{\ge 0}
TeX:
\sum_{k=1}^{m} {k}^{n} = \frac{B_{n + 1}\!\left(m + 1\right) - B_{m + 1}}{m + 1}

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; m \in \mathbb{Z}_{\ge 0}
Definitions:
Fungrim symbol Notation Short description
Sum∑nf(n)\sum_{n} f(n) Sum
Powab{a}^{b} Power
BernoulliPolynomialBn ⁣(z)B_{n}\!\left(z\right) Bernoulli polynomial
BernoulliBBnB_{n} Bernoulli number
ZZGreaterEqualZ≥n\mathbb{Z}_{\ge n} Integers greater than or equal to n
Source code for this entry:
Entry(ID("4aab8a"),
    Formula(Equal(Sum(Pow(k, n), For(k, 1, m)), Div(Sub(BernoulliPolynomial(Add(n, 1), Add(m, 1)), BernoulliB(Add(m, 1))), Add(m, 1)))),
    Variables(n, m),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(m, ZZGreaterEqual(0)))))

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Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC