A Golden (Formal) Power Series

As you probably know, I wear my heart on my sleeve:
Well, I took the golden opportunity (ha!) to bring the golden ratio $\Phi = \frac{1+\sqrt{5}}{2}$ into Calc 2 this week, using it (and its little pal $\Psi = \frac{1-\sqrt{5}}{2}$) to find a closed formula for the $n$-th term of the Fibonacci sequence.
The ubiquitous Fibonacci sequence! It’s something you may have encountered out in the wild. You know, it goes a little like this:
$$F_0 = 1, F_1 = 1, F_n = F_{n-1} + F_{n-2},$$
so that $$F_2 = 2, F_3 = 3, F_4 = 5, F_5 = 8, F_6 = 13, F_7 = 21, \ldots $$
And let’s say for some reason, you need to cook up $F_{108}$. I hope you have some time on your hands if you’re planning to add all the way up to that. Instead, wouldn’t it be nice if we had a simple formula that we could use — i.e., a formula that was not recursive — to figure out the $n$-th Fibonacci number?
Luckily, such a formula exists, and there are lots of ways to find it. In this post, we’ll find it using power series. Read on, brave blogosphere traveler.


Remote Semester Orientation

Advice from Abbi Jutkowitz, Film Editor, who has worked from home on and off for 5+ years, and worked from home exclusively for the past 8 m...