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Taylor's theorem

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This theorem allows us to calculate the error of a Taylor series expansion.

Statement

Let 𝑘≥1 be an integer, and 𝑓:ℝ→ℝ be 𝑘 times differentiable at the point 𝑎∈ℝ. With degree-𝑘 Taylor expansion 𝑃𝑘(𝑥), we have

𝑓(𝑥)=𝑃𝑘(𝑥)+𝑅𝑘(𝑥)

where

𝑅𝑘(𝑥)=𝑓(𝑘+1)(𝑐)𝑘+1(𝑥−𝑎)𝑘+1

for some 𝑎<𝑐<𝑥 (or 𝑥<𝑐<𝑎).

More precise version

Let 𝑘≥1 be an integer, and 𝑓:ℝ→ℝ be 𝑘 times differentiable at the point 𝑎∈ℝ. Then there exists a function ℎ𝑘:ℝ→ℝ such that

𝑓(𝑥)=∑𝑘𝑖=0𝑓(𝑖)(𝑥)𝑖!𝑥𝑖+ℎ𝑘(𝑥)(𝑥−𝑎)𝑘,

and

lim𝑥→𝑎ℎ𝑘(𝑥)=0

References

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