Power Series work in AP Calculus BC.

$\sum_{n=0}^{\infty}(-1)^{n}\frac{x^{2n+1}}{(2n+1)!}=\frac{x}{1!}-\frac{x^3}{3!}+\frac{x^5}{5!}-...$

Process: Since it’s a infinite series, look at partial sums to get an idea what this graph looks like.
So look at
$y_1=\frac{x}{1!}$
$y_2=\frac{x}{1!}-\frac{x^3}{3!}$
$y_2=\frac{x}{1!}-\frac{x^3}{3!}+\frac{x^5}{5!}$

Perfect time to use technology.

Texas Instruments Method

Go to y1. Enter in $y=x$.
Graph.
Wait 3-5 seconds.
Go to y1. Subtract a $\frac{x^3}{3!}$.
Graph.
Wait 3-5 seconds.
Go to y1. Add on a $\frac{x^5}{5!}$.
Graph.
Wait 3-5 seconds.
Go to y1. Subtract a $\frac{x^7}{7!}$.

REALLY crappy resolution. Awful zoom system. Where’s the factorial sign? Hopefully you remember what the previous graphs looked like. Lots of waiting. Ugh.

Desmos: version 1

Graph $y=x$
(NO WAIT STEP)
Subtract a $\frac{x^3}{3!}$.
(STILL NO WAIT STEP)
Add on a $\frac{x^5}{5!}$.
Subtract a $\frac{x^7}{7!}$.

Great!

Desmos: version 2

Students teaching teachers: Have one your students find this out for himself, and remark that they can enter in the entire series, but he’s having trouble finding the infinity sign. SLIDERS!
Link.
Whoa. (That seemed like it should be much harder to type in. Took me 5 minutes to type in the original latex code at the beginning of this post. Took me < 1 minute to actually enter the series into desmos. I had to help 1 student out of 15. That’s it. Wow easy.)

Watch the power series create sin(x) step by step by moving a slider. LIVE!

We live in good times.