log(2)
The integral formulas in http://en.wikipedia.org/wiki/Natural_logarithm_of_2 suggest some variations:
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Integral proof that 2/3<log(2)<3/4
1/2 < log(2) < 1
[2][3]
25/36 > log(2)
From
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the following bounds are obtained for log(2)
An integral and two corresponding (slow) series
Integral:
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Series:
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Integrals to prove that
The following integrals have nonnegative integrand, so the inequalities hold.
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The second integral is in Lucas (2005)
Combining both results,
is obtained.
Simple bounds to prove that
Setting x=0 and x=1 in the denominator of the integral leads to the inequality
Finally,
This remakes the development by Dalzell for , now for this simpler integral related to .
A sequence of integral representations of
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(332/105 is 'almost' the convergent 333/106 but the integrand in this related integral is sign-changing.)
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63/20=3.15
Other integral representations of
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verify(compare to [31] in [4])
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The following general integrals evaluate to the same rational multiples of for nonnegative integer values of n
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Pi/8
Pi/(4n)
Pi/16
From (15) in [5]
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After substituting and simplifying,
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is obtained. Although this integral is equivalent to a six-term series -using a positive basis-, it is actually simpler than (31) in Pi Formulas from Mathworld, which is equivalent to the four-term BBP series.
Integrals involving convergents to Pi
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(333/106 is the third convergent to , see A156618)
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(172/219 is the third convergent to , see A164924)
Following Lucas (2009)
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(there is also the simpler form )
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(as well as the form with numerator )
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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
- [12]
- [13]
- [14]
- [15]
- [16] ( [17])
(TODO: write sums for n>0 instead of n>=0)
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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 in the generalized BBP formula ([19]). Moreover, a series and its corresponding integral can be written for any or . If ,
Setting p=3 and q=1, a series and an integral for the fractional part of is obtained:
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Similarly, p=22 and q=7 yields the series pointed out by Adamchik and Wagon:
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For the third convergent, setting p/q=333/106 yields
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The particular case for the fourth convergent (p=355, q=113) is:
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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
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Particular cases
Setting r=1,
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For the second convergent, gives
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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
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Similarly,
gives
Setting this into the general equation and simplifying,
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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.
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(Lehmer, Am. Math. Monthly 92 (1985) no. 7, p. 449)
(Lehmer, Am. Math. Montly 92 (1985) no. 7, p. 449)
does not converge
Catalan's constant G
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