Taylor Series

Updated 15 Sep 2026

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What it is
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  expresses a function's value near a point as an infinite sum of derivative terms: $f(x_0+h) = \sum_{k=0}^{n} \dfrac{f^{(k)}(x_0)}{k!}h^k + R_n(h)$
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Lagrange remainder
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  $R_n(h) = \dfrac{f^{(n+1)}(\xi)}{(n+1)!}h^{n+1}$ for some $\xi$ between $x_0$ and $x_0+h$ — an upper bound on the truncation error from stopping at order $n$
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worked example: $e^{0.5}$ via order-4 Maclaurin expansion
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  $P_4(0.5) = 1.6484375$ ; true value $e^{0.5} = 1.64872127$ ; actual error $\approx 2.84 \times 10^{-4}$
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  Lagrange bound: $|R_4(0.5)| \leq 7.8125 \times 10^{-4}$ — the actual error falls safely within the bound
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Useful special cases
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  a polynomial of degree $n$ , Taylor-expanded to order $n$ , is represented exactly — the remainder $R_n$ is identically zero
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  the Maclaurin series of $f(x) = \dfrac{1}{1-x}$ is exactly the geometric series $\sum_{k=0}^{\infty} x^k$
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Related
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  Truncation Error

Linked references 3