The M1M1 treasure hunt, week 5
The Adventures of Aless in Wonderland: Chapter 3
Aless was not sorry to bid the queen goodbye and sped off down the road, intrigued by a tatty sign which read "Tea party -- this way."
Soon Aless entered a small café and saw a hare, a hatter and a dormouse sitting around a table.
"No room, no room!" they cried as Aless approached.
"There's plenty of room!" said Aless indignantly and made to sit down at one of the very, very many seats, which strangely were all labelled with numbers between 0 & 2.
"But you can't sit there, in chair number 1.431," hissed the hare. "You have to sit as far away from the Hatter as you can. He's in seat 0.5"
Aless looked at some of the seats nearer the open door which bore the numbers 1.95, 1.96, 1.99, 1.997...
"This is all nonsense!" cried Aless crossly and shut the café door. Now that the room was closed, chair number 2 could be seen and Aless duly sat down on it.
Aless looked round, and muttered, "I must say, this is a most unusual party."
"You mean it is an odd function," corrected the Hatter. "A bit like the 'arm's-length function':
f(y) = ∑n=0 y2n+1/(2n+1) ."
"That is indeed an odd function," replied Aless carefully, by now accustomed to the wordplay of the inhabitants of Wonderland. "But I don't think it is so very unusual."
"You think not?" squeaked the dormouse suddenly. "Then let's work out f(1). To do this, we define another function
g(y) = ∑n=1 y2n/(2n)
Then if you interweave the odd and the even terms, you get
f(1) + g(1) = 1 + 1/2 + 1/3 + 1/4 +...= 2 (1/2 + 1/4 + 1/6 + 1/8 +...) = 2g(1)
So f(1) = g(1). But that can't be right. f(1) is obviously bigger than g(1)." And exhausted by this calculation, the dormouse went back to sleep.
"No, that isn't right," agreed Aless slowly. In fact if I write h(y) = f(y) - g(y) then
"I still don't think the function is so unusual," pouted Aless. "I just don't think it works when y = 1, that's all."
"It's enough to drive you mad!" shouted the hare. "But you haven't had any tea. You may eat anything which the arm's-length function can reach from your seat, which I see is x = 2. You have two arms, so you may only take things from x-values such that f[2(x-2)] 'works,' as you put it, or 'converges' as we say."
"If I ran this café," thought Aless, "People could eat whatever they chose. They would leave singing
'You can get anything you want, in Aless's Restaurant.'
But I suppose I must do as I'm told, so I'll only eat the food at the x-values such that"
"Are all Tea Parties this mad?" cried Aless in frustration.
"Oh, we're relatively sane. There's one in America which..." began the Hatter, but stopped so as not offend anyone. "Why just the other day, one of them decided he would talk all day about not giving medicine to poor people. Of course he began to repeat himself after a while. In fact T hours after starting, the total number of cogent things he had said was:
N(T) = (3T+8T2)/(9+T2) + tanh(T2)/cosh T + [1 - cos(4/T)]log[cosh (T2)] ."
"I do so dislike people who talk too much," mused Aless. "Why, that means he'll only ever say