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  "body": "\n## Statement\n\nLet $a \u003c b$, let $K \\colon [a, b] \\times [a, b] \\to \\C$ be a [[continuous kernel|ZM5G144PZG]] with $M = \\max \\abs{K}$, and let $K_n$ be its [[iterated kernels|9QNBFERMHD]].\nLet $f \\colon [a, b] \\to \\C$ be continuous and let $\\lambda \\in \\C$ with $\\abs{\\lambda} M (b - a) \u003c 1$.\nThen the series\n$$\\varphi(x) = f(x) + \\sum_{n=1}^{\\infty} \\lambda^n \\int_a^b K_n(x, s) f(s) \\, ds$$\nconverges absolutely and uniformly on $[a, b]$.\nIts sum $\\varphi$ is the unique continuous solution of the [[integral equation of the second kind|8KJCN49V64]]\n$$\\varphi(x) = f(x) + \\lambda \\int_a^b K(x, s) \\varphi(s) \\, ds, \\quad x \\in [a, b].$$\n\n## Remarks\n\nProof idea: $\\abs{K_n} \\le M^n (b - a)^{n-1}$, so the $n$-th term is bounded by $\\max \\abs{f} \\cdot (\\abs{\\lambda} M (b - a))^n$, and the [[Weierstrass M-test|0NX5NWAFC9]] gives uniform convergence.\nSubstituting the series into the equation and integrating term by term shows that $\\varphi$ is a solution.\nIf $\\psi$ is a second continuous solution, then $\\max \\abs{\\varphi - \\psi} \\le \\abs{\\lambda} M (b - a) \\max \\abs{\\varphi - \\psi}$, so $\\varphi = \\psi$.\nThis is the [[Neumann series|8QJ91709V2]] $(I - \\lambda K)^{-1} = \\sum_{n \\ge 0} \\lambda^n K^n$ for the integral operator on continuous functions with the maximum norm.\n\nExample: for $K = 1$ on $[0, 1]$ and $\\abs{\\lambda} \u003c 1$, all $K_n = 1$, and the series gives $\\varphi = f + \\frac{\\lambda}{1 - \\lambda} \\int_0^1 f$.\nThe condition on $\\lambda$ cannot be dropped in general: for $\\lambda = 1$ the equation $\\varphi = f + \\int_0^1 \\varphi$ has no solution when $\\int_0^1 f \\ne 0$.\nFor Volterra kernels the bound improves to $\\abs{K_n(x, s)} \\le M^n (x - s)^{n-1} / (n-1)!$, and the series converges for every $\\lambda$.\n\n## Explanations\n\n### L1 human\n\nThe equation says that the unknown is $f$ plus a small correction computed from the unknown itself.\nStart from $f$, compute the correction, add it, and repeat.\nWhen $\\lambda$ is small, each round changes the answer less, and the rounds add up to the exact solution.\n\n### L2 human\n\nThe partial sums of the series are the iterates $\\varphi_0 = f$, $\\varphi_{k} = f + \\lambda K \\varphi_{k-1}$ of the method of successive approximations.\nCollecting the terms gives the resolvent kernel $R = \\sum_{n \\ge 1} \\lambda^{n-1} K_n$ and the formula $\\varphi = f + \\lambda \\int R f$.\nThe series in $\\lambda$ converges for $\\abs{\\lambda}$ below the smallest absolute value of a characteristic value of $K$, and for all $\\lambda$ if there is none.\nBeyond that disc one needs Fredholm determinants or the theory of compact operators.\n",
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