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dan815
 one year ago
expand x/(1xx^2) as a power series about the origin
dan815
 one year ago
expand x/(1xx^2) as a power series about the origin

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ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.2looks suspiciously related to fibonacci

dan815
 one year ago
Best ResponseYou've already chosen the best response.1lol... someone needs to teach him how to hold a pen

Empty
 one year ago
Best ResponseYou've already chosen the best response.2let's say \[f(x)=\sum_{n=0}^\infty a_n x^n\] then this means the difference quotient\[\frac{f(x)f(0)}{x}= \frac{f(x)a_0}{x}=\sum_{n=0}^\infty a_{n+1}x^n\] This is taught at the very beginning of chapter 2 in generatingfunctionology which I'm glad you've just started reading, I'll stop here because I think this is sufficiently interesting. ;P

dan815
 one year ago
Best ResponseYou've already chosen the best response.1u can carry this process for getting powerseries with index shifting higher and higher

dan815
 one year ago
Best ResponseYou've already chosen the best response.1did u get can equation for a powerseiries with k index shifted

dan815
 one year ago
Best ResponseYou've already chosen the best response.1interms of f(x) and f(0)s

Empty
 one year ago
Best ResponseYou've already chosen the best response.2remember you have to dived the whole difference by x, not just the second part

dan815
 one year ago
Best ResponseYou've already chosen the best response.1i didnt work out that pattern, just thinking itd be nice if it was this

dan815
 one year ago
Best ResponseYou've already chosen the best response.1its gotta be this right

dan815
 one year ago
Best ResponseYou've already chosen the best response.1oh ya its kinda trivial hehe

dan815
 one year ago
Best ResponseYou've already chosen the best response.1so what can we do with this

Empty
 one year ago
Best ResponseYou've already chosen the best response.2Well guess you gotta read the book ;P

dan815
 one year ago
Best ResponseYou've already chosen the best response.1will come into stuff like descrete methods of representations for the derivatives

dan815
 one year ago
Best ResponseYou've already chosen the best response.1now that f(0) is popping up all over

dan815
 one year ago
Best ResponseYou've already chosen the best response.1cause for the contiuos version L{f'}=s*L{f}f(0)

dan815
 one year ago
Best ResponseYou've already chosen the best response.1tell me or i will suspend u

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0I would try separating the denominator into partial fractions, then rewriting until the components can be expressed as geometric sums.\[1xx^2=\left(x+\frac{1+\sqrt5}{2}\right)\left(x+\frac{1\sqrt5}{2}\right)\] then \[\frac{x}{1xx^2}=\frac{1\sqrt5}{\sqrt5\left(2x+1\sqrt5\right)}\frac{1+\sqrt5}{\sqrt5\left(2x+1+\sqrt5\right)}\] or something like that...

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Actually, those \(\sqrt5\)s should be \(5\)s in the denominator.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Just to add some more detail, the next step would be to do the following: \[\frac{1}{axb}=\frac{1}{\frac{1}{b}\left(1abx\right)}=b\sum_{k=0}^\infty (abx)^k\]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0$$f(x)=\frac{x}{1xx^2}\\(1xx^2)f(x)=x$$let \(f(x)=\sum_{n=0}^\infty a_n x^n\) so $$f(x)xf(x)x^2 f(x)=x\\\sum_{n=0}^\infty a_n x^n\sum_{n=1}^\infty a_{n1} x^n\sum_{n=2}^\infty a_{n2} x^n=x\\a_0+a_1 xa_0 x+\sum_{n=2}^\infty (a_na_{n1}a_{n2}) x^n=x$$so it follows \(a_n=a_{n1}+a_{n2};\quad a_0=0, a_1=a_0+1=1\)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0ergo \(a_n=F_n\) where \(F_n\) is the nth Fibonacci number where \(F_0=0\) and so $$\frac{x}{1xx^2}=\sum_{n=0}^\infty F_n x^n=x+x^2+2x^3+3x^4+5x^5+\dots$$

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0^^ by the way, this general rule works for all series whose generating functions are rational \(P(x)/Q(x)\)  they are defined by a linear recurrence

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0and generating functions are great (especially in DSP when it comes to ztransforms and analyzing the behavior of systems on digital sequences of data)
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