The maclaurin series (after Colin Maclaurin ) is a term in analysis for the special case of a Taylor series with a development point :
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{\ 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 :
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{\ 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
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{\ displaystyle R_ {n} = {\ frac {x ^ {n + 1}} {(n + 1)!}} f ^ {(n + 1)} (\ theta x) \ qquad 0 <\ theta < 1}
or alternatively
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{\ displaystyle R_ {n} = {\ frac {1} {n!}} \ int \ limits _ {0} ^ {x} (xt) ^ {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
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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.
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Examples
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{\ 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}
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{\ 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 ):
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{\ 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.
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Example: The Taylor series for the natural logarithm function around the expansion point 1, namely
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{\ displaystyle \ ln (x) = \ sum _ {n = 1} ^ {\ infty} {\ frac {(-1) ^ {n-1}} {n}} (x-1) ^ {n} ,}
corresponds to the Maclaurin series
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{\ 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
^ I. Bronstein, K. Semendjajew et al .: Taschenbuch der Mathematik . Verlag Harri Deutsch, Frankfurt am Main 2005, ISBN 3-8171-2006-0 , p. 434.
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