A second take on Taylor series

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Let us start with the observation that

$\int\limits_0^x {f'(x)dx = f(x) - f(0)}$

i.e. that

$f(x) = f(0) + \int\limits_0^x {f'(x)dx}$

We continue with the fact that

$\int\limits_0^x {f''(x)dx = f'(x) - f'(0)}$

$f'(x) = f'(0) + \int\limits_0^x {f''(x)dx}$

Now we substitute that into the expression for f(x) to get

$\begin{gathered} f(x) = f(0) + \int\limits_0^x {\left( {f'(0) + \int\limits_0^x {f''(x)dx} } \right)dx} \hfill \\ \quad \quad = f(0) + f'(0)x + \int\limits_0^x {\int\limits_0^x {f''(x)dxdx} } \hfill \\ \end{gathered}$

We repeat this process with

$f''(x) = f''(0) + \int\limits_0^x {{f^{(3)}}(x)dx}$

This gives us

$\begin{gathered} f(x) = f(0) + f'(0)x + \int\limits_0^x {\int\limits_0^x {f''(x)dxdx} } \hfill \\ \quad \quad = f(0) + f'(0)x + \int\limits_0^x {\int\limits_0^x {(f''(0) + \int\limits_0^x {{f^{(3)}}(x)dx} )dxdx} } \hfill \\ \quad \quad = f(0) + f'(0)x + \frac{{f''(0)}}{2}{x^2} + \int\limits_0^x {\int\limits_0^x {\int\limits_0^x {{f^{(3)}}(x)dx} dxdx} } \hfill \\ \end{gathered}$

Repeating this over and over again using the more general

${f^{(k)}}(x) = {f^{(k)}}(0) + \int\limits_0^x {{f^{(k + 1)}}(x)dx}$

We get

$\displaystyle \begin{array}{l}f(x)=f(0)+{f}'(0)x+\frac{{{f}''(0)}}{2}{{x}^{2}}+\ldots +\frac{{{{f}^{{(n)}}}(0)}}{{n!}}{{x}^{n}}+\ldots \\\quad \quad \quad +\int\limits_{0}^{x}{{\ldots \int\limits_{0}^{x}{{{{f}^{{(n+1)}}}(x)dx\ldots }}dx}}\end{array}$

I.e. the Taylor series expansion with k+1 terms plus some strange integral in the end. On the next page, we will find an upper bound (biggest possible value) of the last term, called the error term.

Up a level : Power Series
Previous page : Proof that ratios of Fibonacci successive numbers tend to the Golden ratio
Next page : Lagrange´s error term

Last modified: Dec 28, 2023 @ 13:53