Today, XKCD, with tongue in cheek, is saying that “The Taylor series should have been canceled after the first term”.

After a flying visit to London (more about that shortly), I am reminded again that at least the second term of the Taylor series impinges on real life.

My house in London is in a courtyard entered through double gates hinged at the outside. By entering a code, you can open either a single or a double gate. The gates are rather slow to open, and of course I get impatient.

Assume for simplicity that the width of each gate is one unit of length, and that the gates open at 1 radian per unit of time. Now, if I open a single gate, then after time *t*, the gap for me to get through is 2 sin *t*/2, whose Taylor series is *t*+…; while if I open both gates, the gap is 2(1−cos t), whose Taylor series is *t*^{2}+…

I have watched people waiting for the double gates to open wide enough for them to slip through, and grumbling about the slowness. Of course I feel a little smug!

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About Peter Cameron

I count all the things that need to be counted.

What you don’t say, is how wide the gap must be for you to pass, and whether that happens in each case at t1.

t < 1, or t > 1

These are slow-moving gates. It would probably take around 20 seconds for them to open one radian. As to the first question, the double gates are wide enough for a car or van to get through (that is twice my unit of length), and I am not very wide. So t is quite small.

Taylor Series for Boolean Functions

Just a little sum thing to play a round with …

Jon