Infinite series for $\pi / 3$ and other identities

INFINITE SERIES FOR π/3 AND OTHER IDENTITIES ROBERT SCHNEIDER Abstract. We combine classical identities for π, log 2, and harmonic numbers, to arrive at a nice infinite series formula for π/3 that does not appear to be well known. In an appendix we give twenty-seven related identities involving π and other irrational numbers. (Note: This is a LaTeX transcription of the author’s first mathematics paper, written in 2006.)

1. Main identity and proof Recall the lovely identity known in the literature as the Gregory–Leibniz formula for π: 1 1 1 1 π = 1 − + − + − ...

(1) 4 3 5 7 9 This identity is immediate from the Maclaurin series expansion of arctan x at x = 1. Here, we prove another simple summation formula that can be used to compute the value of π.

Theorem 1. We have the identity ∞

π X 1 = .

3 n(2n − 1)(4n − 3) n=1 Proof. We begin with the Maclaurin series for the natural logarithm of 2:

∞ X 1 1 1 1 1 (−1)n+1 = lim 1 − + − + −... + − = log 2 (2) n→∞ n 2 3 4 2n − 1 2n n=1 We may rewrite (2) as ∞ X n=1

1 1 − 2n − 1 2n

=

∞ X n=1

1 = log 2, 2n(2n − 1)

thus (3)

∞ X n=1

∞

1 1X 1 1 = = log 2.

4n(4n − 2) 4 n=1 2n(2n − 1) 4

Euler [1] found that the difference between the kth harmonic number and log k approaches a constant γ = 0.5772... (the so-called Euler-Mascheroni constant), so we can write

1 1 1 1 (4) lim 1 + + + + ... + − log(2n) = γ. n→∞ 2 3 4 2n Adding limits from equations (2) and (4), then dividing by 2 (and writing log(2n) = log 2 + log n), we find

1 1 1 1 1 lim 1 + + + ... + − (log 2 + log n) = (log 2 + γ), n→∞ 3 5 2n − 1 2 2 1

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ROBERT SCHNEIDER

thus

(5)

lim

n→∞

1 1 1 1 1 1 + + + ... + − log n = log 2 + γ.

3 5 2n − 1 2 2

Splitting the sum on the left in half and reorganizing, equation (5) may be rewritten

1 1 1 + + ... + (6) lim n→∞ n+1 n+3 2n − 1

1 1 1 1 1 = log 2 + γ − lim 1 + + + ... + − log n . n→∞ 2 3 5 n−1 2 Now, substitute n/2 for n in (5), then subtract 1/2 · log 2 from both sides, to arrive at

1 1 1 1 1 1 log 2 + γ = lim 1 + + + ... + − log n .

(7) n→∞ 2 2 3 5 n−1 2 Adding the corresponding sides of (6) and (7), then subtracting 1/2 · log 2 + γ/2 from the resulting equation, gives

1 1 1 1 (8) lim + + ... + = log 2. n→∞ n+1 n+3 2n − 1 2 Next we will use Leibniz’s formula (1), which we can write in the form

1 1 1 1 1 π (9) lim 1 − + − + −... + − = . n→∞ 3 5 7 2n − 3 2n − 1 4 If we add the limits from (5) and (9), then divide both sides by 2, we find

π 1 1 1 1 1 1 lim 1 + + + ... + − log n = + log 2 + γ . n→∞ 5 9 2n − 3 4 8 2 2 Subtracting 1/2 times the limit in (5) from this equation gives

1 1 1 1 1 1 1 1 1 lim 1 − + − + −... + − − + + ... + n→∞ 2 5 6 2n − 3 2n − 2 2 n+1 n+3 2n − 1 π = .

8 Adding 1/2 times equation (8) to both sides of this expression yields

1 1 π 1 1 1 1 lim 1 − + − + −... + − = + log 2, n→∞ 2 5 6 2n − 3 2n − 2 8 4 which can be rewritten X ∞ ∞ X 1 1 1 π 1 (10) = − = + log 2.

4n − 3 4n − 2 (4n − 2)(4n − 3) 8 4 n=1 n=1 Subtracting equation (3) from (10) yields X ∞ ∞ X 3 1 1 π − = = .

(4n − 2)(4n − 3) 4n(4n − 2) 4n(4n − 2)(4n − 3) 8 n=1 n=1 Finally, multiplication by 8/3 gives the theorem.

INFINITE SERIES FOR π/3 AND OTHER IDENTITIES

3

2. Appendix: Further identities The preceding section is a LaTeX transcription of the author’s first mathematics paper, written in 2006 at a time when I was almost entirely self taught in mathematics. During that period I also found a large number of related summation identities by combining Theorem 1 with other equations in the preceding proof, along with well-known zeta values such as Euler’s famous solution of the Basel problem (see [1]) ∞ X 1 π2 , = n2 6 n=1

and other classical summation identities from [2]. I give a selection of these identities below without proof, loosely organized by number of factors in the denominators of the summands. The first identity follows from the telescoping series (1 − 1/2) + (1/2 − 1/3) + (1/3−1/4)+..., the second and third identities follow by rewriting (2) and (9), respectively, and the rest arise from liberal use of partial fractions. ∞ X

(11)

n=1 ∞ X

(12)

n=1

1 = 2 log 2. (n + 1)(2n + 1)

∞ X

(13)

n=1

(14)

∞ X n=1

n=1 ∞ X

(16)

n=1

(17)

∞ X n=1

(18)

n=1

(19)

n=1

1 π + 6 log 2 = n(4n − 3) 6 1 6 log 2 − π = n(4n − 1) 2

1 log 2 = (2n − 1)(4n − 1)(4n − 3) 2

∞ X

∞ X

π 1 = (4n − 1)(4n − 3) 8

1 π + 2 log 2 = (2n − 1)(4n − 3) 4 ∞ X

(15)

1 1 = (2n − 1)(2n + 1) 2

1 2 log 2 − 1 = n(n + 1)(2n + 1) 2

1 π−2 = (4n + 1)(4n − 1)(4n − 3) 16

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ROBERT SCHNEIDER

∞ X

(20)

n=1 ∞ X

(21)

1 24 log 2 − π 2 = n2 (2n − 1) 6

72 log 2 − 12π − π 2 1 = n2 (4n − 1) 6

n=1

∞ X

(22)

n=1 ∞ X

(23)

n=1

1 π 2 + 24 log 2 − 24 = n2 (2n + 1) 6 ∞ X

(24)

n=1 ∞ X

(25)

n=1 ∞ X

(26)

n=1

(27)

∞ X n=1

1 π2 − 6 = n2 (n + 1) 6 1 π2 − 9 = n2 (n + 1)2 3

π−3 1 = (2n + 1)(2n − 1)(4n + 1)(4n − 1) 6

√ π(1 + 2 2) − 12 1 = (4n + 1)(4n − 1)(8n + 1)(8n − 1) 24 ∞ X

(28)

n=1

1 6 log 2 − π = n(2n − 1)(4n − 1)(4n − 3) 3

∞ X

(29)

n=1 ∞ X

(30)

n=1 ∞ X

(31)

n=1

(32)

1 12 − π 2 = n(n + 1)2 6

∞ X n=1

1 π 2 − 12 log 2 = n2 (n + 1)(2n + 1) 6

1 π 2 + 8π − 24 log 2 = n2 (2n − 1)(4n − 3) 18

1 π 2 + 24π − 120 log 2 = n2 (2n − 1)(4n − 1) 6

1 168 log 2 − 32π − π 2 = n2 (2n − 1)(4n − 1)(4n − 3) 18

INFINITE SERIES FOR π/3 AND OTHER IDENTITIES

∞ X

(33)

n=1 ∞ X

(34)

n=1

52 − 32 log 2 − 27 log 3 1 = n(2n + 1)(2n − 1)(6n + 1)(6n − 1) 16 1 19 + 16 log 2 − 27 log 3 = n(2n + 1)(2n − 1)(3n + 1)(3n − 1) 10 ∞ X

(35)

n=1 ∞ X

(36)

n=1

(37)

∞ X n=1

5

n3 (n

1 = 10 − π 2 3 + 1)

√ 45 − π(3 + 8 2) 1 = (2n + 1)(2n − 1)(4n + 1)(4n − 1)(8n + 1)(8n − 1) 6

1 64 log 2 + 27 log 3 − 74 = n(2n + 1)(2n − 1)(3n + 1)(3n − 1)(6n + 1)(6n − 1) 20 Acknowledgments

I am thankful to Paul Blankenship at Bluegrass Community and Technical College for encouraging my early efforts in analysis; and to David Leep at the University of Kentucky for mentoring me when I had little formal training in mathematics, and prompting me to write up the main result of this study.

References [1] W. Dunham, Euler: the master of us all, The Dolciani Mathematical Expositions, 22. Mathematical Association of America, Washington, DC, 1999.

[2] L. B. W. Jolley, ed. Summation of series. Courier Corporation, 2012.