Partial Sums of Power Series with Fibonacci Coefficients

Theorem

 * $\displaystyle \sum_{k \mathop = 0}^n F_k x^k = \begin{cases}

\dfrac {x^{n + 1} F_{n + 1} + x^{n + 2} F_n - x} {x^2 + x - 1} & : x^2 + x - 1 \ne 0 \\ \dfrac {\left({n + 1}\right) x^n F_{n + 1} + \left({n + 2}\right) x^{n + 1} F_n - 1} {2 x + 1} & : x^2 + x - 1 = 0 \end{cases}$

where $F_n$ denotes the $n$th Fibonacci number.

Proof
Multiplying the summation by $x^2 + x - 1$:

If the denominator is $0$, then $x = \dfrac 1 \phi$ or $x = \dfrac 1 {\hat \phi}$ and the numerator is $0$ also.

Thus we can differentiate the numerator and denominator $x$ and use L'Hôpital's Rule:

Hence the result.