THE GARDEN

Series and numerical differentiation

Updated 11 Oct 2026

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  Taylor approximation

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    Degree-$n$ Taylor polynomial around $a$ -> $P_n(x)=\sum_{k=0}^{n}\frac{f^{(k)}(a)}{k!}(x-a)^k$.

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    Maclaurin series -> Taylor series with $a=0$.

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    Lagrange remainder -> $R_n(x)=\frac{f^{(n+1)}(\xi)(x-a)^{n+1}}{(n+1)!}$ for intermediate $\xi$, under differentiability assumptions.

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    Error bound -> may follow from a derivative bound.

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    smoothness alone does not guarantee the infinite Taylor series equals the function everywhere.

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  Computation

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    $e^x$ -> update successive terms recursively.

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    Sine series -> $t_{k+1}=-\frac{t_kx^2}{(2k+2)(2k+3)}$.

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    Large arguments -> may need range reduction to avoid cancellation and unnecessary work.

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    Geometric series -> $1+x+x^2+\cdots=\frac{1}{1-x}$ only when $|x|<1$.

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  Fourier series

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    Fourier series -> represents periodic behavior using sine and cosine components.

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    Related use: Frequency domain

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    Gibbs phenomenon -> partial sums overshoot and oscillate near a jump discontinuity.

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    adding terms narrows the oscillatory region; limiting overshoot at the jump remains.

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  Finite differences

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    Forward difference -> $\frac{f(x+h)-f(x)}{h}$; truncation error $O(h)$.

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    Central difference -> $\frac{f(x+h)-f(x-h)}{2h}$; truncation error $O(h^2)$ for a sufficiently smooth function.

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    Step size $h$ -> smaller values reduce truncation error but increase roundoff and cancellation.

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    an optimal scale balances these errors.

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  Richardson extrapolation

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    combines estimates at $h$ and $h/2$ to cancel a leading error term when its assumed order applies.

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    it helps only when the function is sufficiently smooth and roundoff has not dominated.

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