1 Euler’s summation formula

∑_{a≤k≤b} f(k) = ∫_{a}^{b} f(x)dx + ∑_{k=1}^{n} \frac{B_{k}}{k!}f^{(k−1)}(x)|_{a}^{b} + R_{m},

where R_{m} = (−1)^{n+1} ∫_{a}^{b} \frac{B_{m}(\{x\})}{m!}f^{(m)}(x)dx

integers a ≤ b, integer n ≥ 1.^{1}

B_{0} = 1 B_{=} − \frac{1}{2} B_{2} = \frac{1}{6} B_{4} = −\frac{1}{30} B_{6} = \frac{1}{42} B_{8} = −\frac{1}{30} B_{3} = B_{5} = … = 0

Bernoulli numbers are defined by an implicit recurrence relation

∑_{j=0}^{m} \binom{m + 1}{j}B_{j} = [m = 0] for all m ≥ 0

Bernoulli polynomial

B_{m}(x) = ∑_{k} \binom{m}{k}B_{k}x^{m−k}

Knuth: Concrete Mathematics

2 Eulerova sumačná formula

Veta 2.1. Nech nezáporná reálna funkcia f má v intervale ⟨0, +∞) spojitú deriváciu. Potom pre každé n = 1, 2, … platí

∑_{k=0}^{n}f(k) = ∫_{0}^{n} f(t)dt + \frac{f(0) + f(n)}{2} + ∫_{0}^{n}(t − [t] − \frac{1}{2})f′(t)dt.

Šalát: Nekonečné rady, s.147

1Ma tam byt k < b ci k < b?