A generalized alternating harmonic series

On Riemann's Rearrangement Theorem for the Alternating Harmonic Series

Let pn be the number of consecutive positive summands and let qn be the number of consecutive negative summands that appear in the classical Riemann’s rearrangement of the alternating harmonic series to sum a prescribed real number s. Assume that s > log 2 and let x = (1/4)e2s. It is shown that the sequence qn is constant equal to 1, and that the values of pn become stabilized: Eventually pn = ...

Sharp Estimates regarding the Remainder of the Alternating Harmonic Series

In the present paper we obtain enhanced estimates regarding the remainder of the alternating harmonic series. More precisely, we show that 1 4n2 +a < ∣ ∣∣ ∣ ∣ n ∑ k=1 (−1)k−1 1 k − (−1)n−1 1 2n − ln2 ∣ ∣∣ ∣ ∣ 1 4n2 +b , for all n ∈N , with a = 2 and b = 2(3−4 ln2) 2 ln2−1 = 1.177398899 . . . . In addition, the constants a and b are the best possible with the above-mentioned property. Mathematic...

A Numerical Integration Method by Using Generalized Series of Functions

Congruences of Alternating Multiple Harmonic Sums

By convention we set H(s;n) = 0 any n < d. We call l(s) := d and |s| := ∑d i=1 |si| its depth and weight, respectively. We point out that l(s) is sometimes called length in the literature. When every si is positive we recover the multiple harmonic sums (MHS for short) whose congruence properties are studied in [9, 10, 17, 18]. There is another “non-strict” version of the AMHS defined as follows...

Congruences involving alternating multiple harmonic sum

We show that for any prime prime p = 2 p−1 k=1 (−1) k k − 1 2 k ≡ − (p−1)/2 k=1 1 k (mod p 3) by expressing the l.h.s. as a combination of alternating multiple harmonic sums.