Where the series comes from

Four lines of algebra, with nothing hidden in them. Every accumulation figure on this site is this derivation evaluated at two numbers somebody typed in.

Educational arithmetic only. Not medical advice and not a dosing protocol.

The derivation

Let one amount be the unit, and let r be the fraction of it still present after exactly one interval:

r = 0.5^{interval ÷ half-life}

Immediately after the first amount, 1 is present. One interval later, r remains, and the second amount brings the total to 1 + r. One interval after that, r(1 + r) = r + r² remains and the third brings it to 1 + r + r². After n amounts:

peak_n = 1 + r + r² + ... + r^n-1

which is a finite geometric series and sums to:

peak_n = (1 - rⁿ) ÷ (1 - r)

Since 0 < r < 1, rⁿ goes to zero as n grows, so the peaks approach 1 ÷ (1 - r) from below and never reach it. The trough just before the next amount is the peak multiplied by r.

The series at r = 0.5, which is what an interval equal to the half-life gives. Every figure here is a simple fraction, which makes this the case to check the arithmetic against by hand. The limit is 2 and it takes 7 intervals to get within one percent of it.
After amount	Peak, as a fraction	Peak	Trough before the next
1	1	1.000	0.500
2	3/2	1.500	0.750
3	7/4	1.750	0.875
4	15/8	1.875	0.938
5	31/16	1.938	0.969
6	63/32	1.969	0.984

What this figure is not

This is educational arithmetic on figures entered by the reader. It is not medical advice, not a dosing protocol, and not a recommendation that anything be prepared, obtained or administered.

Nothing here states what quantity is appropriate for anyone, and there is no function in it that produces an amount to administer. A question about anyone’s health belongs with a licensed clinician who can examine the person asking.

Compounds of the kind this arithmetic is used on are largely research chemicals that are not approved by the FDA for human use. An exact calculation is not an accurate measurement: what is actually in a syringe depends on the preparation and the device, neither of which this page can see.

A half-life belongs to the population, formulation and route it was measured in. Carrying one anywhere else is an assumption rather than a calculation, and this site holds no half-life for any compound: the figure is whatever was typed into the form.

An interval on this site is a parameter, never a proposal. No page here suggests how often anything should happen, and a geometric series evaluated at a spacing somebody chose is not a schedule.

No qualified clinical reviewer is recorded against this site, and no credential is claimed anywhere on it. the full version.

What the derivation assumes

Four things, in order. That the whole of each amount is available immediately, which no subcutaneous injection satisfies. That elimination is first-order at a constant proportional rate, which is an approximation. That each amount is identical to the last. And that nothing about the system changes between one repeat and the next.

Each of those is a modelling assumption rather than arithmetic, and each is printed with every result the worksheet produces, in that order. A figure from this model is a statement about the model.

Why the limit is approached rather than reached

Because rⁿ is never exactly zero. Each repeat closes the remaining gap to the limit by a factor of r, so the approach is geometric: fast at first and then slower without ever finishing. At r = 0.5 the gap halves each time, which is why seven intervals gets within one percent and fourteen gets within a hundredth of one.

Past a certain point the arithmetic saturates for a different reason: in double-precision floating point, 1 - rⁿ rounds to exactly 1 once rⁿ falls below about 1e-16, and the computed peak becomes exactly the limit. That is the hardware being honest rather than a defect, and it is why the test suite for this arithmetic checks monotonicity with a non-strict inequality far out and a strict one early on.

Where the useful reading is

Not in the limit, which is the figure people quote, but in the two columns either side of it: the fraction surviving a gap, which says how much of the previous amount is still in play, and the number of intervals to approach the limit, which says how long the arithmetic takes to settle. A limit of 10 reached after forty intervals and a limit of 2 reached after seven describe completely different situations.

Both figures appear on every page of the half-life and interval families for that reason.