The Universe of Discourse

Sat, 14 Sep 2019

Convergence of Taylor series

[ (Previously) ]

Wikipedia has an animation that shows the convergence of the Maclaurin series for the exponential function:

graphic, which focuses on x between -3 and +3, and y between -5 and
+20.  It displays the curve
y=exp(x) overlaid with its sequence of Maclaurin series
approximations, one after the other.

This makes it look a lot better-behaved than it is. First, because it only shows a narrow region around the center of convergence. But also, it emphasizes the right-hand quadrant, where the convergence is monotone.

If you focus on the left-hand side instead, you see the appoximations thrashing back and forth like a screen door in a hurricane:

The same,
but it focuses on x between -6 and 0, and y between -5 and 1.

This is the same as Wikipedia's animation, just focused on a slightly different region of the plane. But how different it looks!

(Individual frames are available)

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Fare-dodging insurance businesses

A recent article described a business that insured transit riders against being fined for fare-dodging, and asked:

Does anyone remember this? Can someone point me to a reference?

Florian Ilgenfritz wrote to tell me that they had heard of such a system in Berlin, and this let them to the German Wikipedia article on Schwarz­fahrer­ver­sicher­ung, which is precisely what I was looking for. (“Schwarz­fahrer“ is fare-dodging, literally “blackriding”, and “ver­sicher­ung” is insurance.) The article mentions similar systems in Paris and Stockholm, and also one run by the student union of the University of Hanover, which had to shut down because increased fare enforcement on the Hanover metro depleted the benefits pool.

One Paris version, as described at this archived page, was amusingly called le Réseau pour l’Abolition des Transports Payants. (Its initials, RATP, are the same as those of the state-owned public transportation operator). Each RATP group was an insurance cooperative. Would-be fare-dodgers would meet at the beginning of the month and contribute to a pool, from which fines would be reimbursed.

The page says:

  • a group will have 10–30 members
  • each member will put 6–7 Euros into the kitty
  • “even a very unlucky person cannot be fined more than 4 or 5 times a year”

Each member is putting in €72–84 per year. Even if every member is maximally unlucky, this is enough to cover a fine of €16–17, and two or three times that much of not everyone is caught so frequently.

NPR reported on the Stockholm version as recently as 2015, and the Paris version in 2010. There is a German-language article from Der Spiegel from around the same time. This article claims that in one Berlin association, only 50% of the members’ €11 monthly premium had to be paid out in benefits, leaving a €33,000 profit. No word on what happened to the profit. Perhaps it was reinvested for the benefit pool, or perhaps the least scrupulous of the fund organizers embezzled it.

Most Philadelphia public transit is guarded by turnstiles, making fare-dodging more troublesome. The exception is on the regional rail system, where the passenger can board, with or without a ticket, and the conductor comes around to punch tickets, or sell higher-priced on-train tickets to people without any. This is well enough enforced that a fare dodger would be caught almost every time and have to pay the higher price, so the insurance scheme would not be practical. But I remember back in the 1990s it would sometimes be worth while to ride regional rail for a couple of stops in Center City, rather than the subway. The on-board regional rail fare was substantially higher than the subway fare, but the chance of getting off the crowded train without having to buy a ticket was good.

A recent article in the Philadelphia Inquirer describes a different fare-dodging scheme. SEPTA will sell a weekly pass, good for up to 56 rides, for $25.50, or 45.5¢ per ride. The cash fare is $2.50. The fare-dodgers will buy a weekly pass, and then stand at the turnstile offering to let cash-fare riders use it for $2 cash. The cash-fare rider is ahead by $0.50 and the entrepreneur can earn a return of up to 449% on their $22.50 investment.

Exercise for the reader: where is the extra value here and why doesn't SEPTA capture it? Are the entrepreneurs increasing or depleting the public good?

Thanks to M. Ilgenfritz for the initial references, which provided enough for me to get started following up the details on my own.

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