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Taylor series

A Taylor series approximates a function with a polynomial built from the function's derivatives at one point. The more terms we include, the better the approximation near that point.

ex=1+x+x22!+x33!+…e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dotsthe Taylor series of exe^x about x=0x=0

Symbols

n!n!nn factorial

Example

Approximate e0.1e^{0.1} to degree 2: 1+0.1+0.122=1.1051+0.1+\frac{0.1^2}{2}=1.105 (exact value ≈1.1052\approx 1.1052).

Closer to the expansion point gives a better approximation for a given number of terms.
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