Approximating functions with Taylor polynomials and error bounds - Sequence and Series

Approximating functions with Taylor polynomials and error bounds

Lessons

Notes:
To approximate a function with a Taylor Polynomial of n’th degree centred around a=0a=0, use
f(x)f(a)+f(a)(xa)+f"(a)(xa)22!++fn(a)(xa)2n!f(x) \approx f(a) + f'(a)(x-a)+\frac{f^{"}(a)(x-a)^2}{2!}+ \cdots + \frac{f^n (a)(x-a)^2}{n!}

where Pn(x)=f(a)+f(a)(xa)+f"(a)(xa)22!++fn(a)(xa)2n!P_n (x) = f(a)+f'(a)(x-a)+\frac{f^{"}(a)(x-a)^2}{2!}+ \cdots + \frac{f^n (a)(x-a)^2}{n!} is the Taylor Polynomial.

To find the difference between the actual value and your approximated value, look for the error term, which is defined as
Rn(x)=fn+1(z)(xa)n+1(n+1)!R_n(x)=\frac{f^{n+1}(z)(x-a)^{n+1}}{(n+1)!}

Note that adding your Taylor Polynomial with your error would give you the exact value of the function. In other words,
f(x)=Pn(x)+Rn(x)f(x)=P_n(x)+R_n(x)
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Approximating functions with Taylor polynomials and error bounds

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