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User:Jaume Oliver Lafont/Constants
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log(2)
The integral formulas in http://en.wikipedia.org/wiki/Natural_logarithm_of_2 suggest some variations:
verify (following 0.62 and 0.65 in [1])
Integral proof that 2/3<log(2)<3/4
1/2 < log(2) < 1
25/36 > log(2)
From
the following bounds are obtained for log(2)
π
An integral and two corresponding (slow) series
Integral:
Series:
Integrals to prove that 3 < π < 4
The following integrals have nonnegative integrand, so the inequalities hold.
The second integral is I_{1,2} in Lucas (2005)
Combining both results,
is obtained.
Simple bounds to prove that 3 + 1 / 12 < π < 3 + 1 / 6
Setting x=0 and x=1 in the denominator of the integral leads to the inequality
Finally,
This remakes the development by Dalzell for 22 / 7 − π, now for this simpler integral related to π − 3.
A sequence of integral representations of π
verify (332/105 is 'almost' the convergent 333/106 but the integrand in this related integral is signchanging.)
Other integral representations of π
verify(compare to [31] in [4])
The following general integrals evaluate to the same rational multiples of π for nonnegative integer values of n
From (15) in [5]
After substituting and simplifying,
is obtained. Although this integral is equivalent to a sixterm series using a positive basis, it is actually simpler than (31) in Pi Formulas from Mathworld, which is equivalent to the fourterm BBP series.
Integrals involving convergents to Pi
(333/106 is the third convergent to π, see A156618)
(172/219 is the third convergent to π / 4, see A164924)
Following Lucas (2009)
verify (there is also the simpler form I_{1,2})
verify (as well as the form with numerator x^{4}(1 − x)^{4})
Type 3.3 equation (12) in [6] is a linear combination of integrals [7] (larger error) and [8] (smaller error).
An exercise
Given formulas [9] and [10] find an integral for
Write as a linear combination of and :
Split this equation into rational and transcendental parts:
 ,
so
 .
Solve the system to get [11]:
Form the solution as a linear combination of the integrals
and check it.
Series involving convergents to Pi
Using binomial coefficients
(TODO: write sums for n>0 instead of n>=0)
BBP series
Binary series
Adamchik and Wagon note as a curiosity ([18], page 8) that a series for can be written by choosing an appropriate value for r in the generalized BBP formula ([19]). Moreover, a series and its corresponding integral can be written for any c − π or π − c. If c = p / q,
Setting p=3 and q=1, a series and an integral for the fractional part of π is obtained:
Similarly, p=22 and q=7 yields the series pointed out by Adamchik and Wagon:
For the third convergent, setting p/q=333/106 yields
The particular case for the fourth convergent (p=355, q=113) is:
Unfortunately, all these integrands change their sign in (0,1), so the integrals cannot be directly used as a proof that
Slowly convergent series
A general formula
Particular cases
Setting r=1,
For the second convergent, gives
This integrand is not nonnegative. (Find the corresponding integrals for other convergents and check whether they are nonnegative in (0,1) or not; try to find also nonnegative numerators for this denominator (1+x)(1+x^2) by adapting Lucas' algorithm).
For the third convergent 333/106, r=151/106 is the solution of
 ,
so
Similarly,
gives
Setting this r into the general equation and simplifying,
Another general formula
The general formula may be written as a function of p and q in the rational approximation c=p/q
From this formula, particular cases can be directly obtained from the target fraction p/q without computing r.
(log(2))^{2}
π^{2}
(Lehmer, Am. Math. Monthly 92 (1985) no. 7, p. 449)
(Lehmer, Am. Math. Montly 92 (1985) no. 7, p. 449)
does not converge