- #1
Petar Mali
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Homework Statement
How to represent function
[tex]\frac{1}{e^x-x-1}[/tex]
in form of Laurent series around point [tex]0[/tex]
Homework Equations
Laurent series
[tex]f(z)=\sum^{\infty}_{n=-\infty}a_n(z-z_0)^n[/tex]
Here is [tex]z_0=0[/tex]
The Attempt at a Solution
Computer gives
[tex]\frac{2}{x^2}-\frac{2}{3 x}+\frac{1}{18}+\frac{x}{270}-\frac{x^2}{3240}-\frac{x^3}{13608}-\frac{x^4}{2041200}+\frac{x^5}{874800}+\frac{13 x^6}{146966400}-\frac{307 x^7}{24249456000}-\frac{479 x^8}{203695430400}+O[x]^9[/tex]
in form of [tex]12[/tex] first members in series.
[tex]e^{x}=1+x+\frac{x^2}{2!}+...[/tex]
so I can say
[tex]e^x-x-1=\sum^{\infty}_{n=2}\frac{x^n}{n!}[/tex]
[tex]\frac{1}{e^x-x-1}=\frac{1}{\sum^{\infty}_{n=2}\frac{x^n}{n!}}[/tex]
But I don't know what to do with that.