Maclaurin series

The maclaurin series (after Colin Maclaurin ) is a term in analysis for the special case of a Taylor series with a development point : ${\displaystyle x_{0}=0}$

${\displaystyle f(x)=\sum _{j=0}^{\infty }{\frac {f^{(j)}(0)}{j!}}x^{j}=f(0)+f'(0)\cdot x+{\frac {1}{2!}}f''(0)\cdot x^{2}+\dots }$

Considering only finitely many terms in the above series yields Maclaurin's formula as a special case of Taylor's formula :

${\displaystyle f(x)=f(0)+f'(0)\cdot x+{\frac {f''(0)}{2!}}x^{2}+\dots +{\frac {f^{(n)}(0)}{n!}}x^{n}+R_{n}}$

with the remaining link

${\displaystyle R_{n}={\frac {x^{n+1}}{(n+1)!}}f^{(n+1)}(\theta x)\qquad 0<\theta <1}$

or alternatively

${\displaystyle R_{n}={\frac {1}{n!}}\int \limits _{0}^{x}(x-t)^{n}f^{(n+1)}(t)\mathrm {d} t.}$

The convergence of the Maclaurin series can be demonstrated by examining the remainder of the term or by determining the radius of convergence . In the latter case, however, it can happen that the series converges, but its sum is not equal . An example for such a case is the function with the condition : the terms of its Maclaurin series are all 0, but is for ${\displaystyle R_{n}}$ ${\displaystyle f(x)}$ ${\displaystyle f(x)=\exp(-1/x^{2})}$ ${\displaystyle f(0)=0}$ ${\displaystyle f(x)\not =0}$ ${\displaystyle x\not =0.}$

For functions that are not defined at - e.g. B. , or which are defined at, but cannot be differentiated as often as desired - z. B. , also no maclaurin series can be developed. ${\displaystyle x=0}$ ${\displaystyle f(x)={\tfrac {1}{x}}}$ ${\displaystyle x=0}$ ${\displaystyle f(x)=x{\sqrt {x}}}$

Examples

• Sine

${\displaystyle \sin(x)=\sum _{n=0}^{\infty }(-1)^{n}{\frac {x^{2n+1}}{(2n+1)!}}={\frac {x}{1!}}-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-\ldots =x-{\frac {x^{3}}{6}}+{\frac {x^{5}}{120}}-\ldots }$

• Exponential function

${\displaystyle e^{x}=\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}=1+x+{\frac {x^{2}}{2!}}+{\frac {x^{3}}{3!}}+{\frac {x^{4}}{4!}}+\dots =1+x+{\frac {1}{2}}x^{2}+{\frac {1}{6}}x^{3}+{\frac {1}{24}}x^{4}+\dots }$

Conversion of any Taylor series into Maclaurin series

Every Taylor series, including those with a development point , can be understood as a Maclaurin series. For this purpose, instead of the Taylor series, the Taylor series is considered ( substitution ): ${\displaystyle x_{0}\neq 0}$ ${\displaystyle f(x)}$ ${\displaystyle f(x_{0}+x)}$

${\displaystyle f(x_{0}+x)=\sum _{n=0}^{\infty }{\frac {f^{(n)}(x_{0})}{n!}}[(x_{0}+x)-x_{0}]^{n}=\sum _{n=0}^{\infty }{\frac {f^{(n)}(x_{0})}{n!}}x^{n}.}$

By shifting “to the side”, the new development point is currently 0, which means that the new Taylor series is a Maclaurin series. ${\displaystyle -x_{0}}$

Example: The Taylor series for the natural logarithm function around the expansion point 1, namely ${\displaystyle \ln(x)}$

${\displaystyle \ln(x)=\sum _{n=1}^{\infty }{\frac {(-1)^{n-1}}{n}}(x-1)^{n},}$

corresponds to the Maclaurin series ${\displaystyle \ln(x+1).}$

${\displaystyle \ln(x+1)=\sum _{n=1}^{\infty }{\frac {(-1)^{n-1}}{n}}x^{n}=x-{\frac {x^{2}}{2}}+{\frac {x^{3}}{3}}-{\frac {x^{4}}{4}}+\cdots .}$

Individual evidence

1. ^ I. Bronstein, K. Semendjajew et al .: Taschenbuch der Mathematik . Verlag Harri Deutsch, Frankfurt am Main 2005, ISBN 3-8171-2006-0 , p. 434.