Wikipedia:Reference desk/Archives/Mathematics/2022 February 12

= February 12 =

Proofs of rational bounds on π
In Proof that 22/7 exceeds pi a very pretty integral is given:

$$\int_0^1 \frac{x^4\left(1-x\right)^4}{1+x^2} \, dx = \frac{22}{7} - \pi,$$

which is evaluated via rather tedious polynomial long division, but simplifies nicely. $$22/7$$ is one of the continued fraction convergents to $$\pi$$, but the analogous integrals for the next convergents ($$333/106$$, $$355/113$$, etc.) are far more complicated—and they must be, since there must be a factor of 53 and 113 in each antiderivative, respectively. My question is simply whether this is a grand coincidence. The natural generalization (replacing the exponent 4 with $$4n$$, n>1) never seems to give a "good" approximation in Diophantine terms. Cheers, Ovinus (talk) 21:56, 12 February 2022 (UTC)


 * Are you asking whether it is a coincidence that the largest prime factor of the denominators of the next convergents is so much higher? A few steps later, in $$104348/33215,$$ we have $$33215=5\times 7\times 13\times 73,$$ so there is not some rule that the largest prime factor must keep increasing. In the continued-fraction expansion of $$e,$$ we even have a convergent $$87/32.$$ --Lambiam 23:49, 12 February 2022 (UTC)
 * Indeed there's no bound on the convergents' prime factors; I was just noting that for pi in particular, it would be impossible to achieve a value of, for example, $$355/113 - \pi$$ with something like $$\int_0^1P(x)/q(1+x^2)\, dx$$, with $$P$$ being a polynomial with integer coefficients and $$q, \operatorname{deg} P < 112$$. My question was more about intuition for why the long division magically simplifies to such a low common denominator; in the original integral, the antiderivative's coefficients' GCD is $$1/1230$$ or something like that.
 * But as a tangent, your example of $$104348/33215$$ makes me wonder if there are low-degree rational functions that could be used to obtain those approximations too. I'm sure there are, but I don't see a systematic way to find them. Ovinus (talk) 01:24, 13 February 2022 (UTC)
 * You should probably read the Stephen Lucas paper linked to in the article. (Amazingly, you don't have to go through a paywall to get to it.) I think the answer is yes, it is (more or less) a grand coincidence. The Lucas paper does mention that some "experimenting" was needed to produce integrals for 355/113 - pi, which I interpret as meaning that the method used would not be easily generalized. --RDBury (talk) 03:42, 13 February 2022 (UTC)