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Sèries de Taylor i de Laurent típiques

Sèries de Taylor i de Laurent típiques

Sèrie de Taylor general

f(x)=∑n=0∞f(n)(x0)n!(x−x0)n=f(x0)+f′(x0)(x−x0)+f′′(x0)2(x−x0)2⋯f(x)=\sum_{n=0}^\infty \frac{f^{(n)}(x_0)}{n!} (x-x_0)^n \textcolor{#C0C0C0}{=f(x_0)+f'(x_0)(x-x_0)+\frac{f''(x_0)}{2}(x-x_0)^2\cdots}

Sèries de Taylor bàsiques

ex=∑n=0∞xnn!=1+x+x22+x36+x424+x5120⋯e^x=\sum_{n=0}^\infty\frac{x^n}{n!} \textcolor{#C0C0C0}{ = 1+x+\frac{x^2}{2}+\frac{x^3}{6}+\frac{x^4}{24}+\frac{x^5}{120}\cdots}sin⁡x=∑n=0∞(−1)nx2n+1(2n+1)!=x−x36+x5120⋯\sin x=\sum_{n=0}^\infty(-1)^n\frac{x^{2n+1}}{(2n+1)!} \textcolor{#C0C0C0}{ =x-\frac{x^3}{6}+\frac{x^5}{120}\cdots}cos⁡x=∑n=0∞(−1)nx2n(2n)!=1−x22+x424⋯\cos x=\sum_{n=0}^\infty(-1)^n\frac{x^{2n}}{(2n)!} \textcolor{#C0C0C0}{ = 1-\frac{x^2}{2}+\frac{x^4}{24}\cdots}11−x=∑n=0∞xn=1+x+x2+x3⋯\frac{1}{1-x}=\sum_{n=0}^\infty x^n \textcolor{#C0C0C0}{=1+x+x^2+x^3\cdots}ln⁡(1+x)=∑n=0∞(−1)nxn+1n+1=x−x22+x33−x44+⋯\ln(1+x)=\sum_{n=0}^\infty (-1)^n\frac{x^{n+1}}{n+1} \textcolor{#C0C0C0}{=x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\cdots}
Posar-li color al estil
cos⁡x=1−x22+x424⋯=∑n=0∞(−1)nx2n(2n)!\red{\cos x} = 1-\frac{x^2}{2}+\frac{x^4}{24}\cdots = \sum_{n=0}^\infty\red{(-1)^n\frac{x^{2n}}{(2n)!}}
Més series
8.7: Laurent Series
The Laurent series of a complex function f(z) is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases …
8.7: Laurent Series
https://math.libretexts.org/Bookshelves/Analysis/Complex_Variables_with_Applications_(Orloff)/08:_Taylor_and_Laurent_Series/8.07:_Laurent_Series
8.7: Laurent Series