Closed form of the leaky accumulator

Read the equation

Summing the retained contributions q, λq, λ²q, and so on gives q(1−λⁿ)/(1−λ). For |λ|<1 this tends to q/(1−λ). The denominator must be nonzero; at λ=1 the recurrence instead accumulates linearly.

Terms & definitions

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Read the definition from its right-hand expression to the quantity on the left. Its surrounding source records the intended input, state, or parameter context. Where the source leaves an operation unspecified, this page preserves that gap.

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Original context

An+1=ρAn+q,  A0=0,  0<ρ<1,

solving the geometric series gives

An=q1−ρn1−ρ.

By contrast, for an imposed nonleaky rule An+1=An+0.1, threshold two is reached at step 20. That result follows from the nonleakage assumption. It does not establish that all physical or computational loops implement a nonleaky accumulator.

Even positive increments need not produce unbounded accumulation: summable increments such as 2−n have a finite total. A valid crossing theorem needs a condition on the cumulative sum, not just an assertion that data keeps moving.

{A}_{n}=\frac{q\left(1-{\rho }^{n}\right)}{1-\rho }.