Journal of Applied Mathematics and Computational Mechanics 2013, 12(3), 79-91
SOME GENERALIZATIONS OF GREGORY’S POWER SERIES
AND THEIR APPLICATIONS
Natalia Gawrońska, Damian Słota, Roman Wituła, Adam Zielonka
Institute of Mathematics, Silesian University of Technology
Gliwice, Poland
[email protected]
Abstract. The series representing the generalizations of classical James Gregory's series are
discussed in this paper. Formulae describing sums of these series are found. A number
of applications of obtained formulae are also presented, among others, in receiving
the generalizations of Gregory-Leibniz-Nilakantha formula. Moreover, the sums of series
of differences of odd harmonic numbers are generated.
Keywords: Gregory’s series, Gregory-Leibniz-Nilakantha formula
AMS Subject Classification: 40A30, 40A05, 40-04.
Introduction
James Gregory (1638-1675), Scottish mathematician, discovered the Taylor
series more than forty years before Taylor published it. Also the Maclaurin series
for tan , sec , arctan and sec were all known to him, but only the series for
arctan (1671):
arctan = −
ଷ ହ ଻
+ − + ⋯ , || ≤ 1,
3
5
7
(1)
is known as “Gregory’s series” [1]. This series, by applying it into formula
1
1
= 4 arctan − arctan
,
4
5
239
discovered in 1706 by English mathematician John Machin (1680-1752), made it
possible to obtain the relation
=
16
4
4ଶ
4ଷ
1 −
+
−
+ ⋯ −
3 ⋅ 100 5 ⋅ 100ଶ 7 ⋅ 100ଷ
5
4
1
1
−
1 −
+
−⋯
239
3 ⋅ 57121 5 ⋅ 57121
80
N. Gawrońska, D. Słota, R. Wituła, A. Zielonka
used by Machin for numerical calculating the value of (exact to one hundred decimal places) which broke the hegemony of the geometrical method of Archimedes
used from ancient times for determining the approximations of number .
The aim of our paper is to give some generalizations of formula (1). We distinguish an infinite family of such generalizations and we present their various applications, among others, in generating new formulae of numerical nature. In particular,
we give the generalizations of the classic Gregory-Leibniz-Nilakantha formula [2]:
1 1 1
=1− + − +⋯
4
3 5 7
(2)
and their applications for summation of series of differences of odd harmonic
numbers. Formula (2) was historically the first formula in which represented
the sum of numerical series.
The subject matter concerning the summation of the series was already an object
of the authors’ interests, the effect of which is given in paper [3]. In this paper the
Fourier series method, elementary trigonometric and the residue method were used.
For a change, in the current paper we use only the simple calculus method and an
elementary trigonometric one. It appears that in paper [3], as well as in the presented
paper, apart from new results, relations and identities, there are discovered a lot of
inexhaustible, still inspiring subjects which will certainly lead to prepare new works.
1. Generalizations of Gregory’s formula
We begin with determining the generalizations of Gregory alternating power
series (right hand of (1)). Let us take
ଷ ହ ଻
+ − + ⋯,
3
5
7
ଷ
ହ
଻
ଽ ଵଵ ଵଷ ଵହ
ଶ () = + − − + +
−
−
+⋯
3
5
7
9
11
13
15
ଵ () = −
and, in general
ଷ
ଶ௡ିଵ
ଶ௡ାଵ
ଶ௡ାଷ
ସ௡ିଵ
௡ () = + + ⋯ +
−
−
− ⋯−
+
3
2 − 1 2 + 1 2 + 3
4 − 1
ସ௡ାଵ
ସ௡ାଷ
଺௡ିଵ
଺௡ାଵ
଼௡ିଵ
+
+
+⋯+
−
− ⋯−
+⋯
4 + 1 4 + 3
6 − 1 6 + 1
8 − 1
By using the possibility of integrating the power series (by treating every term
separately) in the interval of convergence, we easily get the following relation for
each ∈ [−1,1]:
Some generalizations of Gregory’s power series and their applications
81
௡ () = ( 1 + ଶ + ⋯ + ଶ௡ିଶ − ଶ௡ − ଶ௡ାଶ − ⋯ − ସ௡ିଶ + ସ௡ + ସ௡ାଶ + ⋯
௫
… + ଺௡ିଶ − ଺௡ − ⋯ − ଼௡ିଶ + ଼௡ + ⋯ ) =
଴
= ( 1 + ଶ + ⋯ + ଶ௡ିଶ )(1 − ଶ௡ + ସ௡ − ଺௡ + ⋯ ) =
௫
1 + ଶ + ⋯ + ଶ௡ିଶ
1 − ଶ௡
=
=
=
1 + ଶ௡
(1 − ଶ )(1 + ଶ௡ )
଴
௫
for = 1:
଴
଴
= arctan ,
for = 2:
=
for = 3:
√2
arctan(√2 − 1) + arctan( √2 + 1),
2
1
2
= arctan − arctan ଶ
,
3
3
−1
for = 4:
=
௫
π
π
√2
cos arctan csc − cot + arctan csc + cot +
2
8
8
8
8
8
+ sin arctan sec − tan π + arctan sec + tan π,
8
8
8
8
8
for = 5:
1
2
= arctan + 4 cos
arctan scs − tan + arctan scs + tan +
5
5
5
5
5
5
2
2
2
2
+ 4 cos arctan scs
+ tan + arctan scs
− tan =
5
5
5
5
5
=
1
4 − 10 − 2√5 arctan + √5 − 1 arctan +
5
√5 + 1
4 + 10 − 2√5
+ arctan +
√5 + 1
+√5 + 1 arctan 4 + 10 + 2√5 + arctan 4 − 10 + 2√5,
√5 − 1
√5 − 1
(3)
(4)
(5)
(6)
(7)
82
N. Gawrońska, D. Słota, R. Wituła, A. Zielonka
for = 6:
=
√2
arctan√2 + 1 + arctan√2 − 1 +
6
2√2 − 1 − √3
2√2 + 1 + √3
+√3 + 1 arctan + arctan +
√3 − 1
√3 − 1
2√2 + √3 − 1
2√2 − √3 + 1
+ √3 − 1 arctan + arctan =
√3 + 1
√3 + 1
1
2√2 − 1 − √3 √2
= ଶ +
√3 + 1 arctan +
3
6
√3 − 1
(8)
+arctan 2√2 + 1 + √3 +
√3 − 1
2√2 + √3 − 1
2√2 − √3 + 1
+√3 − 1 arctan + arctan ,
√3 + 1
√3 + 1
for = 7:
1
3
= arctan + 1 + 2 cos − 2 sin − 2 sin ×
7
7
14
14
× arctan sec − tan + arctan sec + tan +
7
7
7
7
3
3
3
+ −1 + 2 cos − 2 sin + 2 sin arctan csc
− cot +
7
14
14
14
14
3
3
+arctan
csc
+ cot +
14
14
3
+ 1 + 2 cos + 2 sin + 2 sin ×
7
14
14
× arctan csc − cot + arctan csc + cot ,
14
14
14
14
for = 9:
(9)
Some generalizations of Gregory’s power series and their applications
2 1
= arctan + arctan2 + √3 + arctan2 − √3 + 9 2
2
+ −1 + cos + cos
− sin ×
9
9
18
× arctan sec − tan + arctan sec + tan +
9
9
9
9
2
+ 1 − cos + cos
− sin ×
9
9
18
2
2
2
2
× arctan sec
− tan + arctan sec
+ tan +
9
9
9
9
2
+ 1 + cos + cos
+ sin ×
9
9
18
× arctan csc − cot + arctan csc + cot .
18
18
18
18
83
(10)
One can see that formulae for = 1,2,3,6 are especially useful for computations.
Formulae for = 4,7,9 are of a more advanced trigonometrical nature, however
they can be used for calculations as well.
Remark 1. Let us notice that computing the values of arctan for > 1 can be
always reduced to computing the values of arctan for ∈ (0,1), since the following identity [4]:
1
√1 + ଶ − 1
arctan = arctan
, > 0
2
is satisfied. Moreover, if we take
() =
√1 + ଶ − 1
, > 0,
then () < which implies that ( ௡ାଵ : = ∘ ௡ , = 1,2, …):
ଵ
ଶ
lim ௡ () = 0
௡→ஶ
for every > 0. Thus, for an appropriately quick computation of arctan values,
it is enough to apply Gregory’s formula for arctan( ௡ ()) for sufficiently big
numbers ∈ ℕ, since
arctan = 2௡ arctan ௡ , > 0.
84
N. Gawrońska, D. Słota, R. Wituła, A. Zielonka
For example
3
1
1
arctan = 2 arctan = 4 arctan
=⋯
4
3
√10 + 3
(11)
Of course, it does not mean that the other formulae, given above, cannot be used.
Quite the contrary, and, what is essential, the other formulae can give even better
convergence.
2. Review of applications of formulae (3)-(10)
We present only the list of selected relations:
a)
√2
arctan√2 − 1 + arctan√2 + 1 =
2
√2
= limష
arctan√2 − 1 + arctan√2 + 1 =
௫→ଵ 2
2√2
√2
= limష
arctan
=
௫→ଵ 2
1 − 2 ଶ − 1
√2
√2
√2
√2
= limష
arctan arctan( ∞) =
,
=
ଶ
௫→ଵ 2
1−
2
4
ଶ 1 =
that is
arctan( √2 − 1) + arctan( √2 + 1) = ,
2
which is a special case of the known formula [2]:
b)
arctan + arctan
1 = sgn, ∈ ℝ\{0}.
2
√2
√2
ଶ =
arctan 2
2
2
which implies the equality
arctan 2 = ( − 1)௞ 4ି௞ ஶ
௞ୀ଴
1
1
+
.
4 + 1 24 + 3
From this and from Gregory’s formula for = ଷ and from the equality, which is
easy to verify [4]:
ଵ
Some generalizations of Gregory’s power series and their applications
3 2 arctan 2 − arctan = ,
4 2
i.e., by (11)
c)
1 arctan 2 − arctan = ,
3 4
85
(12)
we get two series, both of geometrical rate of convergence, linear combination
of which approximates .
ଷ (1) =
2
2
5
− arctan( − ∞) =
− − =
,
12 3
12 3
2
12
which implies the equality
12
1
1
1
=
( − 1)௞ିଵ +
+
.
5
6 − 5 6 − 3 6 − 1
ஶ
(13)
௞ୀଵ
Unfortunately, this alternating series is rather slowly convergent.
d)
1
2
√3
√3 2
√3
√3
ଷ = arctan
+ arctan
=
+ arctan
3
3
18 3
3
3
2
2
which implies
9
1
1
√3
+ 2 arctan
= 3√3 ( − 1)௞ିଵ 27ି௞ +
+
. (14)
6
2
6 − 5 2 − 1 6 − 1
ஶ
௞ୀଵ
We also have
which implies
1
√
√ 2
√
ଷ = arctan
+ arctan
,
3
3
−1
√
√
+ 2 arctan
=
−1
ஶ
ଶ
1
= 3√ ( − 1)௞ିଵ ିଷ௞ +
+
.
6 − 5 6 − 3 6 − 1
arctan
௞ୀଵ
(15)
86
N. Gawrońska, D. Słota, R. Wituła, A. Zielonka
From this, by applying Gregory’s series for arctan ௞, > 1, we can also comଵ
௞
pute the value of arctan ௞ మ ିଵ in the convergence rate of geometrical series.
Moreover, let us notice a curious detail - from equality [4]:
1
1
2
arctan + arctan + arctan = arctan 5,
2
3
3
by applying formula (15) for = 4 and by using Gregory’s formula twice for
ଵ
ଵ
= ଶ and = ଷ, we can calculate arctan 5 in the convergence rate of geometrical series.
1
√2
√2 2
+ arctan √2,
ଷ = arctan
2
2
3
3
e)
which implies
12
2
3
√2
arctan
+ 2 arctan √2 = √2 ( − 1)௞ିଵ 8ି௞ +
+
.
2
6 − 5 2 − 1 6 − 1
ஶ
௞ୀଵ
f)
3 + √3 √2
√2
଺ =
arctan 2 + √3 + 1 arctan−1 + arctan +
2
6
√3 − 1
3 − √3
+√3 − 1 arctan( 1) + arctan =
√3 + 1
=
3 + √3 3 − √3
√2
− + arctan( 2) + √3 + 1arctan +√3 − 1arctan ,
6
2
√3 − 1
√3 + 1
hence, by applying the formulae for a sum and difference of arctangents, the
equality, given below, follows
1
1
3
√3 − + arctan + arctan( 2) − √3arctan2√3 =
3
2
2
(−1)௞
1
1
1
+
+ ଶ
+
଺௞
2
12 + 1 2(12 + 3) 2 (12 + 5)
௞ୀ଴
1
1
1
+
+ ସ
+ ହ
.
ଷ
2 (12 + 7) 2 (12 + 9) 2 (12 + 11)
=
ஶ
(16)
Some generalizations of Gregory’s power series and their applications
87
√2
ସ sin =
sin arctan 2√2 − 1 + 8
2
8
+ cos arctan2 + √2 − arctan√2 =
8
2
√2
=
sin arctan 2√2 − 1 + cos arctan 2
8
8
3 + 2√2
g)
which implies the equality
√2
arctan 2√2 − 1 + √2 + 1arctan 23 − 2√2 =
2
ஶ
(2 − √2)ସ௞
1
2 − √2 = ( − 1)௞
+ ଶ
+
଼௞
2
8 + 1 2 8 + 3
௞ୀ଴
Similarly we get
ଶ
ଷ
+ (2 − √2) + (2 − √2) .
2ସ 8 + 5 2଺ 8 + 7
(17)
√2
ସ cos =
cos arctan 2√2 + 1 +sin arctan2 − √2 + arctan√2 =
8
2
8
8
√2
=
cos arctan 2√2 + 1 + sin arctan 23 + 2√2,
2
8
8
from where we generate the formula in a way “conjugated” with the previous
one
√2
arctan 2√2 + 1 + √2 − 1arctan 23 + 2√2 =
2
ஶ
(2 + √2)ସ௞
1
2 + √2 = ( − 1)௞
+ ଶ
+
଼௞
2
8 + 1 2 8 + 3
௞ୀ଴
(2 + √2)ଶ
(2 + √2)ଷ
+
.
2ସ (8 + 5) 2଺ (8 + 7)
+
In both the last cases we used formulae [5, problem 1.19]:
sin
(18)
1
1
2 ± √2
= 2 − √2, cos = 2 + √2, cos ± sin = ,
8 2
8 2
8
8
2
(19)
tan = √2 − 1, 1 − cot = −√2, 1 + cot = 2 + √2,
8
8
8
−
arctan − arctan = arctan
, whenever > −1.
1 + 88
N. Gawrońska, D. Słota, R. Wituła, A. Zielonka
Remark 2. In Mathematica software the sums of series (16), (17) are presented
with the use of hypergeometric functions which is much less attractive from the
visually-technical point of view.
h)
1
1
4
1 + √5
ହ cos = √5 + 1arctan 1 + +√5 − 1arctan 2 + + arctan =
5
5
4
√5
√5
1 + √5
= ( − 1) 4
ஶ
ଵ଴୬ାଵ
௡
+
1
1
1 + √5 +
+
10 + 1 10 + 3
4
ଶ
1
1 + √5
1
1 + √5
1
1 + √5
+
+
4
4
4
10 + 5
10 + 7
10 + 9
௡ୀ଴
ସ
଺
଼
in which we used relation [5, problem 1.18]:
1 + √5
10 − 2√5
=
⇒ sin =
,
5
4
5
4
2 √5 − 1
2
10 + 2√5
cos
=
⇒ sin
=
.
5
4
5
4
cos
(20)
3. Generalizations of the Gregory-Leibniz-Nilakantha formula
For series (3)-(10) obtained in Section 1 one can apply the Abel’s Limit Theorem
for power series [6, 7]. As a result, we get the set of attractive formulae generalizing
formula (2). For example, from !) we get
1
1
1
+
.
√2 = ( − 1)௞ 4
4 + 1 4 + 3
ஶ
௞ୀ଴
We also receive
3ଷ (1) = limష arctan − 2 arctan
௫→ଵ
i.e.,
3
= + = ,
ଶ − 1
2
2
1
1
1
= ( − 1)௞ +
+
.
2
6 + 1 6 + 3 6 + 5
ஶ
(21)
௞ୀ଴
The next formulae are based on the following auxiliary relations
lim arctan csc " − cot " + arctan csc " + cot " =
2 csc "
2 sin " = limష arctan
=
lim
arctan
= ," ∈ (0, )
ଶ
ଶ
ష
ଶ
௫→ଵ
௫→ଵ
1
−
2
ଶ
1 − csc + cot "
௫→ଵష
Some generalizations of Gregory’s power series and their applications
and
89
lim arctan sec " − tan " + arctan sec " + tan " =
= limష arctan csc − − cot − + arctan csc − + cot − =
௫→ଵ
2
2
2
2
2 cos = limష arctan
= , ∈ (− , ).
௫→ଵ
1 − ଶ
2
2 2
௫→ଵష
Hence, from (6) we deduce
√2
√2
cos ⋅ + sin ⋅ =
cos + sin =
2
4
8 2
8 2
8
8
(ଵଽ) √2
#
2 − √2 + 2 + √2 = 2 + √2
4
8
ସ (1) =
which implies
1
1
1
1
2 + √2 = ( − 1)௞ +
+
+
.
4
8 + 1 8 + 3 8 + 5 8 + 7
ஶ
(22)
௞ୀ଴
Whereas, from (7) we obtain
5ହ 1 =
which implies
2 1
2
+ 4 cos
⋅ + 4 cos ⋅ = 2 + cos + cos =
5 2
5 2
8
5
5
4
(ଶ଴)
1
1 √5
# 2 + = + √5,
4
8 2
1
1
+ √5 = ( − 1)௞ =
5 4
10 + 2$ − 1
ହ
ஶ
∗
∗
,
= ( − 1)௞ %ଵ଴௞ାଽ
− %ଵ଴௞ିଵ
௞ୀ଴
௥ୀଵ
ஶ
(23)
∗
where %ଶ௞ିଵ
denotes the -th odd harmonic number announced in the Introduction
௞ୀ଴
∗
%ଶ௞ିଵ
∶= From (8) we get
3√2଺ 1 =
௞
௟ୀଵ
1
, = 1,2, …
2& − 1
√2
√2
+ √3 = + √3 ,
4
4
90
N. Gawrońska, D. Słota, R. Wituła, A. Zielonka
i.e.,
1
∗
∗
+ √6 = ( − 1)௞ (%ଵଶ௞ାଵଵ
− %ଵଶ௞ିଵ
).
6 2
ஶ
(24)
௞ୀ଴
From (9) we obtain
7଻ (1) =
3
+ 1 + 6 cos − 2 sin + 2 sin 4 2
7
14
14
which gives us the “beautiful” formula
3
3
∗
∗
+ 3 cos − sin + sin = ( − 1)௞ (%ଵସ௞ାଵଷ
− %ଵସ௞ିଵ
).
7 4
7
14
14
ஶ
(25)
௞ୀ଴
At last, from (10) we receive
9
2
ଽ (1) = + 2 + cos + 3 cos
− sin .
2
8 2
9
9
18
Hence, the next our “lovely” formula follows
9
2
∗
∗
+ cos + 3 cos
− sin = ( − 1)௞ (%ଵ଼௞ାଵ଻
− %ଵ଼௞ିଵ
).
9 4
9
9
18
ஶ
(26)
௞ୀ଴
గ
గ
Remark 3. Numbers cos ଻ , sin ଵସ = cos
ଷగ
,
଻
ଷగ
ଶగ
from (25) are line଻
గ
ଶగ
గ
ସగ
cos , cos and sin = cos
ଽ
ଽ
ଵ଼
ଽ
sin ଵସ = cos
arly independent over ℚ. Similarly, numbers
are also linearly independent over ℚ. In both cases, with regard to the appropriate
trigonometric identities, it means that the rational linear combinations of these numbers cannot be rational numbers either. In this connection, it seems to be unlikely to
reduce the number of sums of components on the left sides of formulae (25) and (26).
Final comments. Additional historical remarks concerning Gregory can be
found in [8]. Formulae related to the sums of series, presented in this paper, were
verified with [6], [9], [10] and [11]. The greater part of these formulae seems to be
original, especially original is the method of generating the discussed formulae.
References
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[2] Wituła R., Number ߨ, its history and influence on mathematical creativity, Wyd. Prac. Komputerowej J. Skalmierskiego, Gliwice 2011 (in Polish).
Some generalizations of Gregory’s power series and their applications
[3]
91
Wituła R., Słota D., On the sum of some alternating series, Comput. Math. Appl. 2011, 62,
2658-2664.
[4] Modenov P.S., Collection of Problems for a Special Course of Elementary Mathematics,
Sowietskaja Nauka, Moskva 1957 (in Russian).
[5] Wituła R., Complex Numbers, Polynomials and Partial Fractions Decompositions, Part I, Wyd.
Pol. Śl., Gliwice 2010 (in Polish).
[6] Knopp K., Infinite Series, PWN, Warsaw 1956 (in Polish).
[7] Wituła R., On certain applications of formulae for sum of unimodular complex numbers, Wyd.
Prac. Komputerowej J. Skalmierskiego, Gliwice 2011 (in Polish).
[8] Edwards C.H., The Historical Development of the Calculus, Springer, New York 1979.
[9] Prudnikov A.P., Brychkov Yu.A., Marichev O.I., Integrals and Series, Vol. 1, Elementary Functions, Gordon & Breach Sci. Pub., New York 1986.
[10] Bromwich T.J.I'A., An Introduction to the Theory of Infinite Series, Merchant Books 2008.
[11] Prus-Wiśniowski F., Real Series, Wyd. Nauk. Uniw. Szczecińskiego, Szczecin 2005 (in Polish).