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

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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#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

${\frac {4}{1+t^{2}}}=4-4t^{2}+5t^{4}-4t^{5}+t^{6}-{\frac {t^{4}(1-t)^{4}}{1+t^{2}}}.$

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 < 3959⁄1260 .  --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 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 ${\tfrac {1}{1024}}$".  --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 ([1]), and also [2] are publicly available and presumably cover much of the same ground. --RDBury (talk) 12:44, 5 May 2020 (UTC)