What is sub-linear reciprocal convergence called? Would it be called parabolic convergence? For example, consider the series:
$1,\frac{1}{2},\frac{1}{3},\frac{1}{4},\frac{1}{5},\frac{1}{6},\frac{1}{7},\frac{1}{8}...$
When you sum up such a series, the result gets arbitrarily large. But if you have an alternating sum the result is defined, but convergence is slow.
$\frac{\pi}{4}=1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}-\frac{1}{11}+\frac{1}{13}-\frac{1}{17}...$
Numerical convergence for such an alternating series can be greatly speeded up by using binomial weightings on the last n+1 approximations. My observation is that convergence becomes exponential. What would this be called?
$\sum_{i=0}^{n} S_{m+i} \times 2^{-n} \times {n \choose i}$
For example, if you use the series for Pi, for the first 10 terms you get:
0 +4/1 4.000000000000000000000000000
1 -4/3 2.666666666666666666666666667
2 +4/5 3.466666666666666666666666667
3 -4/7 2.895238095238095238095238095 binomial weightings
4 +4/9 3.339682539682539682539682540 1/64
5 -4/11 2.976046176046176046176046176 6/64
6 +4/13 3.283738483738483738483738484 15/64
7 -4/15 3.017071817071817071817071817 20/64
8 +4/17 3.252365934718875895346483582 15/64
9 -4/19 3.041839618929402211135957266 6/64
10 +4/21 3.232315809405592687326433456 1/64
A binomial weighting for approximations 4..10 gives a much more accurate result
3.141598683394349029023951625