Manuscript Oai276-A@1 · Oai276-A.tex · section · 215068af

Rational sums and explicit remainder bounds

Stages 3 and 6: the explicit finite maximum certificate › Rational sums and explicit remainder bounds · lines 920–941

Oai276-A@1#eq:sums, Oai276-A@1#eq:tails

For $0 \leq z < 1$ , define

L (z) J (z) = k = 0 \sum^{\infty} \frac{z ^{k}}{2 k + 1}, = k = 1 \sum^{\infty} \frac{z ^{k}}{2 k ( 2 k - 1 )}, L_{N} (z) J_{N} (z) = k = 0 \sum^{N - 1} \frac{z ^{k}}{2 k + 1}, = k = 1 \sum^{N} \frac{z ^{k}}{2 k ( 2 k - 1 )} .

Put

T_{N} (z) = \frac{z ^{N}}{( 2 N + 1 ) ( 1 - z )}, R_{N} (z) = \frac{z ^{N + 1}}{( 2 N + 2 ) ( 2 N + 1 ) ( 1 - z )} .

Displayed formulas in this passage

Oai276-A@1#eq:sums · line 923
$L (z) J (z) = k = 0 \sum^{\infty} \frac{z ^{k}}{2 k + 1}, = k = 1 \sum^{\infty} \frac{z ^{k}}{2 k ( 2 k - 1 )}, L_{N} (z) J_{N} (z) = k = 0 \sum^{N - 1} \frac{z ^{k}}{2 k + 1}, = k = 1 \sum^{N} \frac{z ^{k}}{2 k ( 2 k - 1 )} .$
Oai276-A@1#eq:tails · line 934
$T_{N} (z) = \frac{z ^{N}}{( 2 N + 1 ) ( 1 - z )}, R_{N} (z) = \frac{z ^{N + 1}}{( 2 N + 2 ) ( 2 N + 1 ) ( 1 - z )} .$