ExampleConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-21

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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Picard iteration for $y^{'} = y$ , $y (0) = 1$ , recovers the exponential series

Example

For $y^{'} = y$ and $y (0) = 1$ , start from $u_{0} (t) = 1$ . The Picard iterates are

$u_{m} (t) = \sum_{k = 0}^{m} \frac{t^{k}}{k!},$

and converge uniformly on every bounded interval to $\exp t$ , the unique solution.

Facts & Assumptions

Given: The scalar Picard operator $(T u) (t) = 1 + \int_{0}^{t} u (s) d s$ .

[L1]

The iterates $u_{r}$ converge to $\exp$ uniformly on every bounded interval (Picard iteration from $1$ produces the exponential partial sums).

[L2]

Picard-Lindelöf gives a unique local solution of the IVP (Picard-Lindelöf local existence and uniqueness for first-order systems).

Verification

technique · direct

1.1

The cited construction gives $u_{m} (t) = \sum_{k = 0}^{m} t^{k} / k!$ , beginning with $u_{0} = 1$ , and [L1] gives compact-uniform convergence to $\exp$ .

givenL1

2.1

The limit satisfies the Picard equation, and [L2] identifies it with the unique solution of $y^{'} = y$ , $y (0) = 1$ .

step 1.1L2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

Gerald Teschl, Ordinary Differential Equations and Dynamical Systems, Ch. 2 (standard reference, not scraped)
Jiri Lebl, Basic Analysis I, Section 6.3 (standard reference, not scraped)

Picard iteration for y′=yy′=y, y(0)=1y(0)=1, recovers the exponential series