harmonic
harmonic is one function. Everything below came out of it during this
build, at parameters taken from the essays rather than invented for this page — so a figure
here is the same figure a reader meets in an essay, and if the generator changes, this page
changes with it.
At its defaults
show: "blocks"
show: "growth"
show: "overhang"
show: "alternating"
show: "rearranged"
What it checks while it draws
Collected by running the family and listening to lib/verify.js, not written
here. The count is how many separate times this build put that claim to the test.
- block 1 reaches a half ×4
- consecutive partial sums differ by the next term ×1
- each block leans further out than the one below it ×1
- no term is used twice ×1
- the answer is caught between an even and an odd partial sum ×1
- the bound on each block is exactly a half ×1
- the drawn lean is the sum of the drawn steps ×1
- the drawn running total is the partial sum ×1
- the error is smaller than the last term used ×1
- the gap to ln n has settled on γ ×1
- the last term is small ×1
- the plotted partial sums are the series ×1
- the rearranged total settles near the number it was aimed at ×1
- the signs are the ones the series has ×1
- the target is visibly away from the sum the same terms give in order ×1
- the total is not small ×1
Where it is called
Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.
A sum whose terms vanish and whose total does not
Add a half, a third, a quarter, and keep going. The terms shrink to nothing and the total passes every number there is — but so slowly that no computation will ever watch it happen.
AnalysisThe same terms, in a different order, adding to whatever is asked
Flip alternate signs in the harmonic series and it converges. Reorder the terms — add nothing, remove nothing — and it converges to any number chosen in advance. Addition stops being commutative, and the picture shows where it goes.