Expand the following: 1/(1 - x - x^2) and explain the reason it generates the coefficients that it does. Why? I don't understand it. Thanks for help.
+9665 \(\frac{1}{1-x-x^2}\\\)Step 1
\(= \frac{1}{(x-\frac{1+\sqrt5}{2})(x-\frac{1-\sqrt5}{2})}\)Step 2
\(=\frac{x+\frac{1+\sqrt5}{2}}{(x-\frac{1+\sqrt5}{2})(x+\frac{1+\sqrt5}{2})}\times \frac{x+\frac{1-\sqrt{5}}{2}}{(x-\frac{1-\sqrt5}{2})(x+\frac{1-\sqrt{5}}{2})}\)Step 3
\(=\frac{2x+1+\sqrt5}{2x^2-(3+\sqrt{5})}\times \frac{2x+1-\sqrt{5}}{2x^2-(3-\sqrt{5})}\)Step 4
\(=\frac{((2x+1)+\sqrt5)((2x+1)-\sqrt5)}{((2x^2-3)-\sqrt{5})(((2x^2-3)+\sqrt{5}))}\)Step 5
\(=\frac{(2x+1)^2-{\sqrt5}^2}{(2x^2-3)^2-{\sqrt5}^2}\)Step 6
\(=\frac{4x^2+4x+1-5}{4x^4-12x^2+9-5}\)Step 7
\(=\frac{4x^2+4x-4}{4x^4-12x^2+4}\)Step 8
\(=\frac{x^2+x-4}{x^4-3x^2+1}\)
The coefficient is generated because the denominator of the 2 fractions in step 3 involves a fraction with denominator 2. So that 2 is multiplied in step 4 so the coefficient is generated.
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MaxWong: Your "solution" is more confusing than non-solution!. That is NOT how you expand a generating series:
Series expansion at x=0:
1+x+2 x^2+3 x^3+5 x^4+8 x^5+13 x^6+21 x^7+34 x^8+55 x^9+89 x^10+...........etc.
(Taylor series)
(converges when abs(x)<1/2 (-1+sqrt(5))),
The coefficients form:1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89..........looks familiar?? Fibonacci sequence!.
But, I don't know "Why".
