Wronski’s Formula For Pi: A First Limit


Previously:

When I last looked at Józef Maria Hoëne-Wronski’s attempted definition of π I had gotten it to this. Take the function:

f(x) = -2 x 2^{\frac{1}{2}\cdot \frac{1}{x}} \sin\left(\frac{\pi}{4}\cdot \frac{1}{x}\right)

And find its limit when ‘x’ is ∞. Formally, you want to do this by proving there’s some number, let’s say ‘L’. And ‘L’ has the property that you can pick any margin-of-error number ε that’s bigger than zero. And whatever that ε is, there’s some number ‘N’ so that whenever ‘x’ is bigger than ‘N’, ‘f(x)’ is larger than ‘L – ε’ and also smaller than ‘L + ε’. This can be a lot of mucking about with expressions to prove.

Fortunately we have shortcuts. There’s work we can do that gets us ‘L’, and we can rely on other proofs that show that this must be the limit of ‘f(x)’ at some value ‘a’. I use ‘a’ because that doesn’t commit me to talking about ∞ or any other particular value. The first approach is to just evaluate ‘f(a)’. If you get something meaningful, great! We’re done. That’s the limit of ‘f(x)’ at ‘a’. This approach is called “substitution” — you’re substituting ‘a’ for ‘x’ in the expression of ‘f(x)’ — and it’s great. Except that if your problem’s interesting then substitution won’t work. Still, maybe Wronski’s formula turns out to be lucky. Fit in ∞ where ‘x’ appears and we get:

f(\infty) = -2 \infty 2^{\frac{1}{2}\cdot \frac{1}{\infty}} \sin\left(\frac{\pi}{4}\cdot \frac{1}{\infty}\right)

So … all right. Not quite there yet. But we can get there. For example, \frac{1}{\infty} has to be — well. It’s what you would expect if you were a kid and not worried about rigor: 0. We can make it rigorous if you like. (It goes like this: Pick any ε larger than 0. Then whenever ‘x’ is larger than \frac{1}{\epsilon} then \frac{1}{x} is less than ε. So the limit of \frac{1}{x} at ∞ has to be 0.) So let’s run with this: replace all those \frac{1}{\infty} expressions with 0. Then we’ve got:

f(\infty) = -2 \infty 2^{0} \sin\left(0\right)

The sine of 0 is 0. 20 is 1. So substitution tells us limit is -2 times ∞ times 1 times 0. That there’s an ∞ in there isn’t a problem. A limit can be infinitely large. Think of the limit of ‘x2‘ at ∞. An infinitely large thing times an infinitely large thing is fine. The limit of ‘x ex‘ at ∞ is infinitely large. A zero times a zero is fine; that’s zero again. But having an ∞ times a 0? That’s trouble. ∞ times something should be huge; anything times zero should be 0; which term wins?

So we have to fall back on alternate plans. Fortunately there’s a tool we have for limits when we’d otherwise have to face an infinitely large thing times a zero.

I hope to write about this next time. I apologize for not getting through it today but time wouldn’t let me.

Author: Joseph Nebus

I was born 198 years to the day after Johnny Appleseed. The differences between us do not end there. He/him.

2 thoughts on “Wronski’s Formula For Pi: A First Limit”

  1. Oh….this is a beautiful “series of pie-sized” articles on “pi” that you are writing. I hope to peruse these series and make some notes for myself. Especially, since I had heard that an Indian immortal math genius, Srinivasa Ramanujan had given an algorithm to compute the value of pi uptil more than a million digits after decimal point.( But, I have always been afraid to read even elementary literature related to this area… ) And, further that such algorithms of Ramanujan are used now to test the efficiency and efficacy of supercomputers…:-)

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    1. Thanks kindly, and I hope that you enjoy.

      Ramanujan had many outstanding formulas, a fair number of them for π. I’m not sure that I know any of them well enough that I could explain how to use them, though, or why any should be so. Wronski’s formula here is interesting because the thing didn’t convince anyone and doesn’t quite parse. But I think it can be made meaningful and be interesting along the way.

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