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After T. Piezas answered Is there a series to show $22\pi^4>2143\,$? a natural question is

Is there a series that proves $\frac{22}{7}-\pi>0$?

One such series may be found combining linearly the series that arise from truncating $$\sum_{k=0}^\infty \frac{48}{(4k+3)(4k+5)(4k+7)(4k+9)} = \frac{16}{5}-\pi$$ to two and three terms, namely

$$\sum_{k=2}^\infty \frac{48}{(4 k+3) (4 k+5) (4 k+7) (4 k+9)} = \frac{141616}{45045}-\pi$$ and $$\sum_{k=3}^\infty \frac{48}{(4 k+3) (4 k+5) (4 k+7) (4 k+9)} = \frac{2406464}{765765}-\pi$$ Solving $$a\left(\frac{141616}{45045}-\pi\right)+b\left(\frac{2406464}{765765}-\pi\right)=\frac{22}{7}-\pi$$ for rational $a,b$ and some algebra manipulation yields the result

$$\frac{16}{21} \sum_{k=0}^\infty \frac{1008 k^2+6952 k+12625}{(4 k+11) (4 k+13) (4 k+15) (4 k+17) (4 k+19) (4 k+21)}=\frac{22}{7}-\pi$$

It is interesting to note that the coefficients needed to multiply the two component series are both positive $$a=\frac{113}{7·8·9}$$ $$b=\frac{391}{7·8·9}$$

because the truncation points have been chosen so that

$$\frac{2406464}{765765}<\frac{22}{7}<\frac{141616}{45045}$$

This procedure yields a result that proves the claim with no need for further processing, and it is readily seen to prove $\frac{p}{q}-\pi>0$ for all fractions between $\pi$ and $\frac{16}{5}$.

Now, in the light of this equivalent form of Lehmer's formula $$\pi-3=\sum_{k=1}^\infty \frac{4!}{(4k+1)(4k+2)(4k+4)}$$

one may still ask

Q1 Is there a series that proves $\frac{22}{7}-\pi>0$ with constant numerator?

Q2 Is there a reason why $113$ is both the numerator of the $a$ coefficient and the denominator of the next convergent from above $\frac{355}{113}$?

Edit: A similar series with smaller coefficients may be obtained by applying the method above to $$\begin{align} \sum_{k=0}^\infty \frac{960}{(4 k+3) (4 k+5) (4 k+7) (4 k+9) (4 k+11) (4 k+13)} &= \frac{992}{315}-\pi \\ &= \frac{3·333-7}{3·106-3}-\pi \\ \end{align} $$ in order to obtain $$\sum_{k=0}^\infty \frac{96 (160 k^2+422 k+405)}{(4 k+3) (4 k+5) (4 k+7) (4 k+9) (4 k+11) (4 k+13) (4 k+15) (4 k+17)} = \frac{22}{7}-\pi$$

Q3 What is the relationship between $\frac{992}{315}$ and the third convergent to $\pi$ $\frac{333}{106}$?

Community
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Jaume Oliver Lafont
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3 Answers3

12

Q1

Evaluating the following series $$\begin{align} &\sum_{k=0}^\infty \frac{240}{(4k+5)(4k+6)(4k+7)(4k+9)(4k+10)(4k+11)} \\ &= \sum_{k=0}^\infty \left(\frac{1}{4k+5}-\frac{4}{4k+6}+\frac{5}{4k+7}-\frac{5}{4k+9}+\frac{4}{4k+10}-\frac{1}{4k+11}\right) \\ &= \sum_{k=0}^\infty \int_{0}^1\left(x^{4k+4}-4x^{4k+5}+5x^{4k+6}-5x^{4k+8}+4x^{4k+9}-x^{4k+10}\right)dx \\ &= \int_{0}^1 x^4\sum_{k=0}^\infty \left(x^{4k}-4x^{4k+1}+5x^{4k+2}-5x^{4k+4}+4x^{4k+5}-x^{4k+6}\right)dx \\ &= \int_{0}^1 x^4\frac{1-4x+5x^2-5x^4+4x^5-x^6}{1-x^4}dx \\ &= \int_{0}^1 x^4\frac{(1-x^2)(1-x)^4}{(1-x^2)(1+x^2)}dx=\int_{0}^1 \frac{x^4(1-x)^4}{1+x^2}dx=\frac{22}{7}-\pi \\ \end{align}$$ shows its connection with Dalzell's integral.

This may be rewritten as $$\sum_{k=1}^\infty \frac{240}{(4k+1)(4k+2)(4k+3)(4k+5)(4k+6)(4k+7)}=\frac{22}{7}-\pi$$

which appears in the 2009 document by Peter Bala New series for old functions http://oeis.org/A002117/a002117.pdf (formula 5.1) and shows that $\frac{22}{7}-\pi$ can be obtained by taking one term out of the summation in the series $$\sum_{k=0}^\infty \frac{240}{(4k+1)(4k+2)(4k+3)(4k+5)(4k+6)(4k+7)}=\frac{10}{3}-\pi$$

Consecutive truncations yield the inequality

$$\pi...<\frac{141514}{45045}<\frac{10886}{3465}<\frac{22}{7}<\frac{10}{3}$$

Similar fractions, but now converging to $\pi$ from below, may be obtained from the series

$$\sum_{k=0}^\infty \frac{240}{(4 k+3) (4 k+4) (4 k+5) (4 k+7) (4 k+8) (4 k+9)} = \pi-\frac{47}{15}$$

This yields

$$\frac{47}{15}<\frac{1979}{630}<\frac{141511}{45045}<\frac{9622853}{3063060}<...\pi$$

(See a similar inequality for $\log(2)$)

Correspondence between series and integrals

$$\sum_{k=n}^\infty \frac{240}{(4k+1)(4k+2)(4k+3)(4k+5)(4k+6)(4k+7)}=\int_0^1 \frac{x^{4n}(1-x)^4}{1+x^2}dx$$

$$\sum_{k=n}^\infty \frac{240}{(4 k+3) (4 k+4) (4 k+5) (4 k+7) (4 k+8) (4 k+9)}=\int_0^1 \frac{x^{4n+2}(1-x)^4}{1+x^2}dx$$

Equivalent expressions

The general term for these series may be written in compact form using factorials, binomial coefficients or the Beta integral $B$ (see this comment by N. Elkies).

$$\begin{align} \frac{22}{7}-\pi &= 3840\sum_{k=1}^\infty \frac{(k+2)!(4k)!}{(4k+8)!k!} \\ \\ &= \frac{4}{21} \sum_{k=1}^\infty \frac{\displaystyle{k+2 \choose 2}}{\displaystyle{4k+8\choose 8}} \\ \\ &= \frac{4}{21} \sum_{k=1}^\infty \frac{k+1}{\displaystyle{4k+7\choose 7}} \\ \\ &= \frac{16}{21} \sum_{k=1}^\infty \frac{B(4k+1,8)}{B(k+1,2)} \end{align} $$

Interpretation of $\frac{22}{7}-\pi$

Similar series and approximations

If we use the Pochhammer symbol to express this series: $$\sum_{k=0}^\infty \frac{7!(k+1)}{(4k+1)_7}=\frac{7}{4}(10-3\pi)\approx 1$$

we can change the numbers to obtain variants such as

$$ \sum_{k=0}^\infty \frac{5!(k+1)}{(3k+1)_{5}} = \frac{5}{9}\left(2\sqrt{3}\pi-9\right)\approx 1, $$

$$\sum_{k=0}^\infty \frac{11! (k+1)}{(6 k+1)_{11}} = 231-\frac{4565 \pi}{36 \sqrt{3}}\approx 1$$

and $$\sum_{k=0}^\infty \frac{15!(k+1)}{(8k+1)_{15}}=\frac{15}{8}(1716-7(99\sqrt{2}-62)\pi)\approx 1$$

Given that all three series evaluate to almost 1, the following corresponding approximations are derived

$$\begin{align} \pi &=\frac{9\sqrt{3}}{5}+\sqrt{3}\int_0^1\frac{x^3(1-x)^2(1+x)}{1+x+x^2}dx\\ &\approx\frac{9\sqrt{3}}{5} \\ \pi &=\frac{1656\sqrt{3}}{913}- \frac{6\sqrt{3}}{83}\int_0^1 \frac{x^6(1-x)^8}{1+x^2+x^4} dx\\ &\approx\frac{1656\sqrt{3}}{913} \\ \pi &=\frac{1838 \left(62 + 99 \sqrt{2}\right)}{118185}-\frac{62+99\sqrt{2}}{15758}\int_0^1 \frac{x^8(1-x)^{12}}{1+x^2+x^4+x^6}dx\\ &\approx \frac{1838 \left(62 + 99 \sqrt{2}\right)}{118185} \end{align}$$

which give 1, 5 and 8 correct decimals respectively.

The fraction $\frac{1838}{118185}$ is the eighth convergent of $\frac{\pi}{62+99\sqrt{2}}$

Another series and integral for $\frac{22}{7}-\pi$

$$\begin{align} &\sum_{k=0}^\infty \frac{285120}{(4k+2)(4k+3)(4k+5)(4k+6)(4k+7)(4k+9)(4k+10)(4k+11)(4k+13)(4k+14)} \\ &= \frac{1}{28}\int_{0}^1 \frac{x(1-x)^8(2+7x+2x^2)}{1+x^2}dx=\frac{22}{7}-\pi \\ \end{align}$$

Jaume Oliver Lafont
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  • So based on several series versions already found for 22/7 - Pi, it appears that similar to infinite number of integral expressions (as it was shown by Thomas Baruchel), there are infinite number of series as well. – Alex Feb 15 '16 at 23:20
  • You are quite wrong wishing ALL the terms be positive. Anyway your example is nice. Regards. – Piquito Feb 16 '16 at 11:58
  • @Piquito Reading the proof in reverse shows how the positive integrand naturally leads to all terms positive. – Jaume Oliver Lafont Feb 17 '16 at 10:18
1

Proof that $\frac{22}{7}$ exceeds $\pi$.

$$0<\int_0^1\frac{x^4(1-x)^4}{1+x^2}dx=\frac{22}{7}-\pi$$

Proof-

$$\int_0^1\frac{x^4(1-x)^4}{1+x^2}dx$$

$$=\int_0^1x^6-4x^5+5x^4-4x^2+4-\frac{4}{1+x^2}dx$$

$$=\frac {x^7}{7}+\frac{2x^6}{3}+x^5-\frac{4x^3}{3}+4x-4\tan^{-1}(x)\vert_0^1$$

Now,by applying $\tan^{-1}1=45^\circ=\frac\pi4$ and substituting it in the integral and solving the integral yields $\frac{22}{7}-\pi$

orion
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Soham
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  • Why is that integral greater than zero ? Is is just because integral represents an area ? Moreover, what is the usual strategy to evaluate this integral ? – Saikat Feb 13 '16 at 05:13
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    @tatan: since this does not answer either question, would you please write it as a comment? – Jaume Oliver Lafont Feb 13 '16 at 05:18
  • @JaumeOliverLafont Its also a series.... – Soham Feb 13 '16 at 05:27
  • @user230452 I have added the proof.... – Soham Feb 13 '16 at 05:28
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    @tatan your answer is about the integral proof for $\frac{22}{7}-\pi>0$, while this question addresses proofs using series with positive terms only. – Jaume Oliver Lafont Feb 13 '16 at 05:34
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    @user230452 It's $>0$ because the integrand is always $>0$ in the domain of integration. – YoTengoUnLCD Feb 13 '16 at 05:40
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    I have no prove of it, but $\displaystyle \sum_{k=0}^{+\infty}\dfrac{41760+576k}{(4k+3)(4k+5)(4k+7)(4k+9)(4k+11)(4k+13)(4k+15)(4k+17)}=\dfrac{22}{7}-\pi$ – FDP Feb 14 '16 at 19:26
  • @FDP Great! Your numerator is $288(145+2k)$ WA check http://www.wolframalpha.com/input/?i=sum(k%3D0..inf,(41760%2B576k)%2F((4k%2B3)(4k%2B5)(4k%2B7)(4k%2B9)(4k%2B11)(4k%2B13)(4k%2B15)(4k%2B17))) Partial fraction expansion and then grouping into Gregory-Leibniz pairs can prove it by hand, if you want to. Would you write this into an answer? If not, at least as a comment to the question, not to this answer, please. – Jaume Oliver Lafont Feb 14 '16 at 20:29
  • @FDP How did you get it? Combining three truncations instead of two? If that is the case, possibly combining four lets impose one more condition, namely, cancelling the $k$ term? – Jaume Oliver Lafont Feb 14 '16 at 20:45
  • It seems such equalities aren't rare. I got this one experimenting with lindep function of PARI GP (LLL stuff). – FDP Feb 14 '16 at 21:02
  • This is a Riemann integral so it is a limit of series and it is obvious positive. On the other hand, it is unnecessary that ALL the termes are positive as Olivert Lafont wishes (he is wrong in this). – Piquito Feb 16 '16 at 11:56
  • @Piquito Am I correct? – Soham Feb 16 '16 at 17:42
  • @tatan: I think implicitly yes. However it would be nice to add the idea of Riemann integrale – Piquito Feb 17 '16 at 01:09
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Let $\sum_{k=0}^\infty a_n$ any series converging to $\pi$ and choose any series converging to $\frac{22}{7}$, for instance $\sum_{k=0}^\infty \left(\frac{15}{22}\right)^n$

No problem to show that $$\sum_{k=0}^\infty \left(\left(\frac{15}{22}\right)^n -a_n\right)=\frac{22}{7}-\pi\gt 0 $$

CiaPan
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Piquito
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  • Does this series have all terms positive? – Jaume Oliver Lafont Feb 15 '16 at 23:42
  • From a certain range, yes. – Piquito Feb 15 '16 at 23:48
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    How can we prove $\frac{22}{7}-\pi>0$ from that series if not all terms are positive? – Jaume Oliver Lafont Feb 15 '16 at 23:54
  • Theorem.-Let $a=\sum a_n$ and $b=\sum b_n$ two convergent series. Then , for all pair of constants $\alpha$, $\beta$ the series $\sum(\alpha a_n+\beta b_n)$ converges to $\alpha a+ \beta b$ (Mathematical Analysis, T. M. Apostol). Can you exhibit an example of what you say? – Piquito Feb 16 '16 at 00:46
  • This proves $$\sum_{k=0}^\infty \left((\frac{15}{22})^n -a_n\right)=\frac{22}{7}-\pi$$ but not $$\frac{22}{7}-\pi>0$$ – Jaume Oliver Lafont Feb 16 '16 at 00:51
  • ► I understand your game now (something like show that 2> 1). Anyway there are countless examples like yours and none for which the result be $\le 0$. I assumed that, since the number $\frac{22}{7}=3.142....$ and $\pi=3.141....$, we have $\sum (a_n-b_n)$ is positive from a certain index n (you can take as the first term of the series the corresponding to this n). I have no desire to continue. I stop here. Regards. – Piquito Feb 16 '16 at 11:31