The M1M1 treasure hunt, week 4
The Adventures of Aless in Wonderland: Chapter 2
Aless was still worried about last week's function.
"An infinite series," thought Aless, "is like a very long road. But no matter how far I travel down it I've still got a very very long way to go. In which case, what is the point of starting?"
So Aless set off through Wonderland to find someone who could explain infinity. After a while, a hookah-smoking caterpillar came into view.
"Please sir," asked Aless politely. "You have almost an infinite number of legs. How do you ever manage to get anywhere?"
"I most certainly do not," puffed the caterpillar pompously. "I have a good number of legs. More than you, as it happens. When I take a step, my first pair of legs take 1/2 a second, as do the second pair, and then because I'm stretched like a rubber band my n'th pair of legs only need n/2n seconds to move forward. They catch up very quickly."
"Why," exclaimed Aless. "That's a bit like a Geometric Series!"
"It IS a geometric series," snapped the caterpillar. "If you consult a good dictionary, you will see that a "geometer" is a caterpillar, or earth-measurer. So anything of mine is geometric."
"But I don't know how many legs you have!" cried Aless.
"It hardly matters, does it? I have enough, and each step takes me in total about
"But are you saying the tail of the series doesn't matter?" worried Aless.
"Not to me. If you care about tails, you should talk to the Cheshire Cat over there."
Aless had always liked cats, and so set off hurriedly and stopped near a loud purring noise. But the cat was nowhere to be seen.
"Hello?" called Aless. "Can you explain when infinite tails matter?"
Suddenly a cat's head appeared out of nowhere. "Now that is hard to say," grinned the cat. "Sometimes only my head matters, like now. Sometimes my body is the weighty part; sometimes I need part of my tail also, and sometimes just my whiskers are enough. And of course, sometimes the entire series vanishes."
During this speech, the cat emphasised her points by making various parts of her body appear and disappear and finally vanished with a flourish.
"Please stop doing that!" cried Aless crossly. "I need to know when I can stop writing, or I'll have to go on for ever!"
"Well, what I do," confided the cat, "is to label bits of my body. When x is very small, I call my head O(1), my body O(x), and then bits of my tail O(x2), O(x3) and so on. Then I can just include O(xn) in my answer which means "everything the same as or smaller than xn." Even the Queen lets me get away with that."
"So how does that work?" pondered Aless. "I suppose I expand as an infinite series in x and wrap the tail into an O, like:"
"That's easy enough. By the way, did you say the Queen? I'd awfully like to meet her."
"Would you really?" muttered the Cheshire Cat. "She has a bit of a temper. The important thing to remember is to keep your head. She's over there, playing croquet."
Aless continued down the road towards a strange gathering.
"Off with his head!" screamed an imposing red figure, whom Aless immediately identified as the Queen, as she pointed at a quivering gardener.
"Please, your majesty," asked Aless bravely, "what has he done wrong?"
"He painted my white roses red. At least, he probably did. Let the chance of his painting a rose red given that it was white be P1 and the chance of
his painting a red rose red be P2. Then the probability the rose was originally white given that he has with probability P3 painted it red, is, by the theorem, um, that proves that... OFF WITH HIS HEAD! I'll work it out later."
"You can't execute him before he's found guilty!" protested Aless.
"Yes I can. As Queen I may commute sentences. And if the sentence and the verdict commute I can do them in either order. Don't you know anything, child?"
"But...that's not what commute means - it means you can change the general terms of a sentence..." began Aless.
"Oh very well," said the queen. "Your sentence is commuted to the
general term of this series:"