MATHEMATICAL ANALYSIS 10
Taylor's Theorem
Taylor's theorem approximates a sufficiently smooth function near a point using its derivatives there—and tells us about the error.
Taylor polynomial
About \(x=a\), the degree-\(n\) Taylor polynomial is
\[P_n(x)=\sum_{k=0}^{n}\frac{f^{(k)}(a)}{k!}(x-a)^k.\]
Taylor's theorem
Under suitable differentiability assumptions,
\[\boxed{f(x)=P_n(x)+R_n(x)}\]where the Lagrange remainder has the form
\[R_n(x)=\frac{f^{(n+1)}(\xi)}{(n+1)!}(x-a)^{n+1}\]for some \(\xi\) between \(a\) and \(x\).
Example: exponential
\[e^x\approx1+x+\frac{x^2}{2}+\frac{x^3}{6}\]near \(x=0\).
Why the remainder matters
A polynomial approximation alone is not a theorem about accuracy. The remainder quantifies the difference between the function and its approximation.