How to find the full Taylor expansion of the following:

Question

I need to find the full Taylor expansion of $$f(x)=\frac{1+x}{1-2x-x^2}$$

Any help would be appreciated. I'd prefer hints/advice before a full answer is given. I have tried to do partial fractions\reductions. I separated the two in hopes of finding a known geometric sum but I could not.

Edit: I guess you could say that I did not have the.... insight to take the path with the partial decomposition mentioned. I have done some work (I had to go to the gym that is why it took a while)

$$\frac{1+x}{1-2x-x^2}=\frac{1}{2(\sqrt{2}-x-1)}-\frac{1}{2(\sqrt{2}+x+1)}$$ I am going to work with this to go further.

I got to this:

$$\frac{1}{2}\left(\sum_{n=0}^\infty\frac{x^n}{(\sqrt{2}-1)^{n+1}}+\sum_{n=0}^\infty\frac{x^n}{(-\sqrt{2}-1)^{n+1}}\right) $$ But I think this is wrong for some reason.

Edit: Figured it out.

$$\begin{align*} \implies\frac{1+x}{1-2x-x^2}&=\frac{1}{2(\sqrt{2}-x-1)}-\frac{1}{2(\sqrt{2}+x+1)} \\[2mm] &=\frac{1}{2}\left(\frac{1}{a-x}-\frac{1}{b+x}\right) \mbox{where $a=\sqrt{2}-1$ and $b=\sqrt{2}+1$}. \\[2mm] &=\frac{1}{2}\left(\frac{1}{a} \frac{1}{1-\frac{x}{a}}-\frac{1}{b} \frac{1}{1-\frac{x}{-b}}\right) \\[2mm] &=\frac{1}{2}\left(\frac{1}{a}\sum_{n=0}^\infty \frac{1}{a^n}x^n-\frac{1}{b}\sum_{n=0}^\infty\frac{1}{(-b)^n}x^n\right) \\[2mm] &=\frac{1}{2}\left(\frac{1}{\sqrt{2}-1}\sum_{n=0}^\infty \frac{1}{(\sqrt{2}-1)^n}x^n-\frac{1}{\sqrt{2}+1}\sum_{n=0}^\infty\frac{1}{(-\sqrt{2}-1)^n}x^n\right) \\[2mm] &=\frac{1}{2}\left(\sum_{n=0}^\infty\frac{x^n}{(\sqrt{2}-1)^{n+1}}+\sum_{n=0}^\infty\frac{x^n}{(-\sqrt{2}-1)^{n+1}}\right) \\ &=1+3x+7x^2+17x^3+\ldots \end{align*}$$

Answer 1

Hint: $1 - 2x - x^2 = 2 - (1 + 2x + x^2) = 2 - (x + 1)^2 = (\sqrt{2} - (x + 1))(\sqrt{2} + (x + 1))$

Answer 2

Hint For the function $$g(x) = \frac{1}{1-x} \text{ over } x \in (-1,1),$$ the Taylor series is the ordinary geometric series $$\frac{1}{1-x} = 1 + x + x^2 + \ldots = \sum_{k=0}^\infty x^k.$$

If the partial fraction decomposition is given by $$ \frac{A}{x-B} + \frac{C}{x-D} $$ how can you get the decomposition you are seeking?

Answer 3

Here is an alternate approach. Suppose that $$ \frac{1+x}{1-2x-x^2}=\sum_{k=0}^\infty a_kx^k $$ Multiply by $1-2x-x^2$ to get $$ \begin{align} 1+x &=\underbrace{a_0}_{1}+\underbrace{(a_1-2a_0)}_1x+\sum_{k=2}^\infty\underbrace{(a_k-2a_{k-1}-a_{k-2})}_{0}x^k \end{align} $$ Equating coefficients of $x^k$ gives $$ a_0=1,a_1=3,\text{ and }a_k=2a_{k-1}+a_{k-2}\text{ for }k\ge2 $$ Therefore, $$ \frac{1+x}{1-2x-x^2}=1+3x+7x^2+17x^3+41x^4+99x^5+\dots $$

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If we wish a formula for $a_n$ we can solve the recurrence $$ a_k=2a_{k-1}+a_{k-2} $$ using the roots of $x^2-2x-1=0$, which are $1\pm\sqrt2$, and $a_0=1,a_1=3$ to get $$ a_k=\frac{(1+\sqrt2)^{k+1}+(1-\sqrt2)^{k+1}}{2} $$ which is the integer closest to $\dfrac{(1+\sqrt2)^{k+1}}{2}$.