Deret Taylor: Perbedaan antara revisi

[[Berkas:Exp series.gif|ka|jmpl|[[Fungsi eksponensial]] (warna biru), dan jumlahan suku ke ''n''+1 awal deret Taylornya di titik 0 and (warna merah).]]

 

 

== Definisi ==

:<math> \sum_{n=0} ^ {\infin } \frac {f^{(n)}(a)}{n!} \, (x-a)^{n}</math>

 

 

 

==Kesalahan perkiraan dan konvergensi==

Kesalahan dalam perkiraan tersebut tidak lebih dari nilai {{math|{{sfrac|{{mabs|''x''}}⁹|9!}}}}. Secara khusus, untuk nilai {{math|−1 < ''x'' < 1}}, kesalahannya kurang dari&nbsp;0.000003.

 

 

 

:<math>

f(x) = \begin{cases}

\end{cases}

</math>

 

In [[real analysis]], this example shows that there are [[infinitely differentiable function]]s {{math|''f''&thinsp;(''x'')}} whose Taylor series are ''not'' equal to {{math|''f''&thinsp;(''x'')}} even if they converge. By contrast, the [[holomorphic function]]s studied in [[complex analysis]] always possess a convergent Taylor series, and even the Taylor series of [[meromorphic function]]s, which might have singularities, never converge to a value different from the function itself. The complex function {{math|''e''^−1/''z''²}}, however, does not approach 0 when {{mvar|z}} approaches 0 along the imaginary axis, so it is not [[Continuous function|continuous]] in the complex plane and its Taylor series is undefined at 0.-->

 

 

<!--A function cannot be written as a Taylor series centred at a [[singularity (mathematics)|singularity]]; in these cases, one can often still achieve a series expansion if one allows also negative powers of the variable {{mvar|x}}; see [[Laurent series]]. For example, {{math|''f''&thinsp;(''x'') {{=}} ''e''^−1/''x''²}} can be written as a Laurent series.-->

 

==Generalisasi==

:<math>\lim_{h\to 0^+}\sum_{n=0}^\infty \frac{t^n}{n!}\frac{\Delta_h^nf(a)}{h^n} = f(a+t).</math>

 

:<math>\sum_{n=0}^\infty\frac{u^n}{n!}\Delta^na_i = e^{-u}\sum_{j=0}^\infty\frac{u^j}{j!}a_{i+j}.</math>

Jadi secara khusus,

<!--The series on the right is the [[expectation value]] of {{math|''f''&thinsp;(''a'' + ''X'')}}, where {{mvar|X}} is a [[Poisson distribution|Poisson-distributed]] [[random variable]] that takes the value {{math|''jh''}} with probability {{math|''e''^{−''t''/''h''}·{{sfrac|(''t''/''h''){{isup|''j''}}|''j''!}}}}. Hence,

:<math>f(a+t) = \lim_{h\to 0^+} \int_{-\infty}^\infty f(a+x)dP_{\frac{t}{h},h}(x).</math>-->

 

{{see also|Daftar deret matematika}}

 

=== Fungsi eksponensial ===

[[Berkas:Exp series.gif|right|thumb|[[Fungsi eksponensial]] {{math|''e''^''x''}} (berwarna biru), dan jumlah dari yang pertama {{math|''n'' + 1}} persyaratan seri Taylor-nya di 0 (merah).]]

:<math>e^{x} = \sum^{\infty}_{n=0} \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots </math>.

 

=== Logaritma natural ===

:<math>\begin{align}

\ln(1-x) &= - \sum^{\infty}_{n=1} \frac{x^n}n = -x - \frac{x^2}2 - \frac{x^3}3 - \cdots , \\

\ln(1+x) &= \sum^\infty_{n=1} (-1)^{n+1}\frac{x^n}n = x - \frac{x^2}2 + \frac{x^3}3 - \cdots .

\end{align}</math>

 

 

 

:<math>\begin{align}

\frac{1}{(1-x)^3} &= \sum^\infty_{n=2} \frac{(n-1)n}{2} x^{n-2}.

\end{align}</math>

 

 

 

:<math>(1+x)^\alpha = \sum_{n=0}^\infty \binom{\alpha}{n} x^n</math>

 

 

: <math>\binom{\alpha}{n} = \prod_{k=1}^n \frac{\alpha-k+1}k = \frac{\alpha(\alpha-1)\cdots(\alpha-n+1)}{n!}.</math>

 

 

 

:<math>\begin{align}

\end{align}</math>

 

 

== Perkiraan dalam fungsi ==

[[Fungsi trigonometri]] biasa dan inversnya memiliki deret Maclaurin berikut:

:<math>\begin{align}

\end{align}</math>

 

 

=== Fungsi hiperbolik ===

\end{align}</math>

 

 

==Deret Taylor sebagai definisi==

{{sect-stub}}

<!--Taylor series are used to define functions and "[[operator (mathematics)|operator]]s" in diverse areas of mathematics. In particular, this is true in areas where the classical definitions of functions break down. For example, using Taylor series, one may extend analytic functions to sets of matrices and operators, such as the [[matrix exponential]] or [[matrix logarithm]].

 

\end{align}</math>

 

 

:<math>f(a,b) +(x-a) f_x(a,b) +(y-b) f_y(a,b) + \frac{1}{2!}\Big( (x-a)^2 f_{xx}(a,b) + 2(x-a)(y-b) f_{xy}(a,b) +(y-b)^2 f_{yy}(a,b) \Big)</math>

:<math>T(\mathbf{x}) = f(\mathbf{a}) + (\mathbf{x} - \mathbf{a})^\mathsf{T} D f(\mathbf{a}) + \frac{1}{2!} (\mathbf{x} - \mathbf{a})^\mathsf{T} \left \{D^2 f(\mathbf{a}) \right \} (\mathbf{x} - \mathbf{a}) + \cdots,</math>

 

 

:<math>T(\mathbf{x}) = \sum_{|\alpha| \geq 0}\frac{(\mathbf{x}-\mathbf{a})^\alpha}{\alpha !} \left({\mathrm{\partial}^{\alpha}}f\right)(\mathbf{a}),</math>

 

 

=== Contoh ===

\end{align}</math>

 

:<math>e^x\ln(1+y)= y + xy - \frac{y^2}{2} + \cdots, \qquad |y| < 1.</math>

 

* {{MathWorld| urlname= TaylorSeries| title= Taylor Series}}

* [http://math.fullerton.edu/mathews/c2003/TaylorSeriesMod.html Taylor Series Representation Module by John H. Mathews]

* [[minea.org]]

* [http://stud3.tuwien.ac.at/~e0004876/taylor/Taylor_en.html Another Taylor visualisation] {{Webarchive|url=https://web.archive.org/web/20070605020930/http://stud3.tuwien.ac.at/~e0004876/taylor/Taylor_en.html |date=2007-06-05 }} - where you can choose the point of the approximation and the number of derivatives

 

{{Deret (matematika)}}

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