Wikipedia:Reference desk/Archives/Mathematics/2020 May 4
Appearance
Mathematics desk | ||
---|---|---|
< May 3 | << Apr | May | Jun >> | May 5 > |
Welcome to the Wikipedia Mathematics Reference Desk Archives |
---|
The page you are currently viewing is a transcluded archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages. |
May 4
[edit]1935 paper on 22/7 - π
[edit]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#The proof, 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 • 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:
The first published statement of this result was in 1971 by Dalzell [3], although anecdotal evidence [2] suggests it was known by Kurt Mahler in the mid-1960s.
- 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 • 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 < π < 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 223⁄71 < π < 22⁄7 is due to Archimedes. In the article Dalzell proves sharper bounds. He starts by observing that
- Definite integration of both sides then yields
- The remaining integral is bounded by
- So 1979⁄630 < π < 3959⁄1260 . --Lambiam 18:32, 4 May 2020 (UTC)
- Extending this idea a bit further you can get the series expansion:
- where the numbers 1, 21, 126, 462 are the binomial coefficients C(2*n+5,5). (See OEIS: A002299.) 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 1979⁄630 < 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 ". --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 ([1]), and also [2] are publicly available and presumably cover much of the same ground. --RDBury (talk) 12:44, 5 May 2020 (UTC)
- Extending this idea a bit further you can get the series expansion:
- The inequality 223⁄71 < π < 22⁄7 is due to Archimedes. In the article Dalzell proves sharper bounds. He starts by observing that