Read the equation
Each update retains a fraction λ or ρ of the previous value and adds q. With initial value zero and 0≤λ<1, past inputs decay and the state approaches q/(1−λ). This is a standard finite-capacity accumulation example whose long-term behavior follows from a geometric series.
Terms & definitions
| Term | Meaning in this source |
|---|---|
| A_n | Accumulated state |
| lambda_ | Retention factor; ρ in another version |
| q | Constant input |
How it works
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
Read the surrounding source passage
7.2 Proposition 5: repeated positive input does not always accumulate without bound For a leaky accumulator, An+1=ρAn+q, A0=0, 0<ρ<1, solving the geometric series gives An=q1−ρn1−ρ.
Exact original notation
{A}_{n+1}=\rho {A}_{n}+q, {A}_{0}=0, 0<\rho <1,Sources & provenance
{A}_{n+1}=\rho {A}_{n}+q, {A}_{0}=0, 0<\rho <1,
{A}_{n+1}=\rho {A}_{n}+q, {A}_{0}=0, 0<\rho <1,