Wikipedia:Reference desk/Archives/Mathematics/2020 May 4

= May 4 =

1935 paper on 22/7 - π
Can anyone give me the precise reference for a 1935 paper in (something like) the London Mathematical Journal, where the value of a small definite integral from 0 to 1 was 22/7 - π? →2A00:23C6:AA08:E500:CDA3:11DC:85FA:EC73 (talk) 10:45, 4 May 2020 (UTC)


 * See Proof that 22/7 exceeds π, where it gives the origin as a problem in the 1968 Putnam exam. There's a source for that claim, which I haven't checked, so I don't know if the result is actually older or not. –Deacon Vorbis (carbon &bull; videos) 12:34, 4 May 2020 (UTC)
 * On second glance, the article doesn't really claim that this is the origin, but the cited article by Lucas states:
 * Although, our article even has a reference to Dalzell from 1944 instead of 1971: Dalzell, D. P. (1944), "On 22/7", Journal of the London Mathematical Society, 19 (75 Part 3): 133–134. That might be what you were thinking of after all. –Deacon Vorbis (carbon &bull; videos) 12:56, 4 May 2020 (UTC)
 * Yes, that's the one, many thanks. I seem to recall that the paper showed the double-sided inequality $223/71$ < $\pi$ < $22/7$ by a second integral, but can't remember the details. >2A00:23C6:AA08:E500:418A:A36C:4641:8B78 (talk) 13:15, 4 May 2020 (UTC)
 * Yes, that's the one, many thanks. I seem to recall that the paper showed the double-sided inequality $223/71$ < $\pi$ < $22/7$ by a second integral, but can't remember the details. >2A00:23C6:AA08:E500:418A:A36C:4641:8B78 (talk) 13:15, 4 May 2020 (UTC)


 * The inequality $1979/630$ < π < $3959/1260$ is due to Archimedes. In the article Dalzell proves sharper bounds. He starts by observing that
 * $$\frac{4}{1+t^2} = 4 - 4t^2 + 5t^4 - 4t^5 + t^6 - \frac{t^4(1-t)^4}{1+t^2}.$$
 * Definite integration of both sides then yields
 * $$\pi = \frac{22}{7}- \int_0^1 \frac{t^4(1-t)^4}{1+t^2} dt.$$
 * The remaining integral is bounded by
 * $$\frac{1}{1260} = \frac{1}{2} \int_0^1 t^4(1-t)^4 dt < \int_0^1 \frac{t^4(1-t)^4}{1+t^2} dt < \int_0^1 t^4(1-t)^4 dt = \frac{1}{630}.$$
 * So $1979/630$ < π < ⇭⇭⇭. --Lambiam 18:32, 4 May 2020 (UTC)
 * Extending this idea a bit further you can get the series expansion:
 * $$\pi = 10/3 - \frac{1}{5\cdot 1} + \frac{1}{5\cdot 21} - \frac{1}{5\cdot 126} + \frac{1}{5\cdot 462} - \dots$$
 * where the numbers 1, 21, 126, 462 are the binomial coefficients C(2*n+5,5). (See .) Stopping at the third term gives π<22/7. The above series can also be derived from the Leibniz formula for π using various series manipulations. Continuing this you can get more rapidly converging series in terms of C(2*n+k,k) for any odd k. --RDBury (talk) 23:24, 4 May 2020 (UTC)
 * And stopping at the fourth term gives Darzell's lower bound ⇭⇭⇭ < π. --Lambiam 11:42, 5 May 2020 (UTC)
 * In the paper, Darzell also extends the idea in a different way to obtain a convergent series whose terms are more complicated but that has faster convergence; Darzell writes that the terms "are less in magnitude than those of a geometric series of common ratio $$\tfrac{1}{1024}$$". --Lambiam 06:56, 5 May 2020 (UTC)
 * The Darzell papers are behind pay walls, but the S.K. Lucas paper cited in the article mentioned above, and also are publicly available and presumably cover much of the same ground. --RDBury (talk) 12:44, 5 May 2020 (UTC)