Here’s a puzzle based on a passing remark in Andrew Hodges’ biography of Alan Turing.

Hodges says that in 1927, Alan Turing, then 15 years old, discovered the Taylor series for arctangent “starting from the trigonometrical formula for tan(x/2).”

Deriving the series could be a homework exercise in a calculus class, but Turing discovered the series without using calculus.

Hodge doesn’t give any further information about what Turing’s derivation. What might it have been? How could you derive the series expansion for arctangent from a trig identity?

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Seems pretty straightforward to me. Set:

arctan x = a0 + a1 x + a2 x^2 + …

The formula that we’ll use is the Weierstrass transformation:

tan 2θ = 2 tan θ / (1 – tan² θ)

Substitute θ = arctan x to get:

2 arctan x = arctan [2x / (1-x²)]

Finally, replace arctan with its power series on both sides, and solve for a0, a1, a2 and so on.

You’ll need a power series for 2x / (1-x²), which if you’re a 15-year-old Alan Turing, you can probably do by solving:

2x = (1-x²) (b0 + b1 x + b2 x^2 + …)

to obtain:

2x / (1-x²) = 2x + 2x^3 + 2x^5 + 2x^7 + …

To save typing and/or writing, we can also use the fact that arctan is odd, and hence a0 = a2 = a4 = … = 0.

Sorry, that should read “Weierstrass substitution”. The Weierstrass transform is something else entirely.

Pseudonym — nice. I was close but missed a couple crucial steps!

Actually it’s quite possible that a very bright 15 year old at a public (i.e. private) school, Sherborne in Turing’s case, might have been taking calculus

Calculus was almost certainly in the mathematics syllabus at Turing’s school, as it was in the 1980s when I was at secondary school. Elementary differential and integral calculus (usually presented in a theorem-‘lite’ way) is routinely part of the high school curriculum in many countries from the age of 15 or 16. Of course, it’s still quite possible Turing derived this with no calculus as Pseudonym suggests.

I don’t have the book with me, but I believe the author said that Turing had not taken calculus. Of course Turing may have known a little calculus before having taken it.

Let’s suppose Turing didn’t know calculus, or at least not enough to know about series. In that case he — or some hypothetical person in such a position, to move away from historical details — would not start by assuming an infinite series exists and then solve for the values of the coefficients. Instead, he would come up with a recurrence relation and iterate that.

John, I think you’re assuming that you need to study calculus before you see infinite series. You need calculus before you can fully understand it, I agree, but plenty of mathematically-inclined teenagers (myself included), have discovered infinite series before calculus.

One place where they turn up, for example, is geometric series. Another is generating functions.

I find it difficult to believe that even a teenager as bright as Turing, would be able to derive an infinite series

if they didn’t even know that infinite series existed. It seems more likely that the teenage Turing had previously heard of infinite series than that he independently invented the concept.I tried Pseudonym’s solution and ran into the following problem. When the two power series expressions are equated to allow for solving for the coefficients, the result looks like:

2 a1 x + 2 a3 x^3 +2 a5 x^5 +…= 2 a1 x + (2 a1 + 8 a3) x^3 + (2 a1 + 24 a3 +32 a5) x^5 + …

The problem is the first order term, which leaves the solution indeterminate since there is always one fewer independent equation than there are variables.

Did I make a mistake?