SUMS OF ARCTANGENTS AND SOME FORMULAE OF
RAMANUJAN
GEORGE BOROS AND VICTOR H. MOLL
Abstract. We present diverse methods to evaluate arctangent and related
sums.
1. Introduction
The evaluation of arctangent sums of the form
(1.1)
∞
X
tan−1 h(k)
k=1
for a rational function h reappear in the literature from time to time. For instance
the evaluation of
∞
X
2
3π
(1.2)
tan−1 2 =
k
4
k=1
is completely elementary and was proposed by J. Anglesio [1] in 1993. This is a
classical problem that appears in [7, 9, 13], among other places. The evaluation of
√
√
∞
X
−1 1
−1 tan(π/ 2) − tanh(π/ 2)
√
√
tan
(1.3)
= tan
k2
tan(π/ 2) + tanh(π/ 2)
k=1
is somewhat less elementary. R.J. Chapman [6] proposed in 1990 to evaluate this
sum in closed form (in terms of a finite number of trigonometric and hyperbolic
functions). This was solved by A. Sarkar [15] using the techniques described in
Section 3.
The goal of this paper is to discuss the evaluation of these sums and the related
rational sums
n
X
(1.4)
S(n) =
R(k)
k=1
−1
for a rational function R. Note that tan
x will always denote the principal value.
We make use of the addition formula for tan−1 x:
(
x+y
tan−1 1−xy
−1
−1
(1.5)
tan x + tan y =
x+y
tan−1 1−xy
+ π sign x
Date: March 23, 2001.
1991 Mathematics Subject Classification. Primary 33.
Key words and phrases. Arctangent, Series.
1
if xy < 1
if xy > 1
2
GEORGE BOROS AND VICTOR H. MOLL
and
tan−1 x + tan−1
(1.6)
1
x
=
π
sign x.
2
2. The method of telescoping
The closed-form evaluation of a finite sum
n
X
S(n) :=
ak
k=1
is elementary if one can find a sequence {bk } such that
ak
= bk − bk−1 .
In this situation, the sum S(n) telescopes, i.e.
S(n) :=
n
X
ak =
k=1
n
X
bk − bk−1 = bn − b0 .
k=1
This method can be extended to situations in which the telescoping nature of ak is
hidden by a function.
Theorem 2.1. Let f (x) be of fixed sign and define h by
h(x)
(2.1)
=
f (x + 1) − f (x)
.
1 + f (x + 1)f (x)
Then
n
X
(2.2)
tan−1 h(k) =
tan−1 f (n + 1) − tan−1 f (1).
k=1
In particular, if f has a limit at ∞ (including the possibility of f (∞) = ∞), then
∞
X
(2.3)
tan−1 h(k) =
tan−1 f (∞) − tan−1 f (1).
k=1
Proof. Since
tan−1 h(k)
= tan−1 f (k + 1) − tan−1 f (k),
(2.2) follows by telescoping.
Note. The hypothesis on the sign of f is included in order to avoid the case xy > 1
in (1.5). In general, (2.2) has to be replaced by
(2.4)
n
X
k=1
tan−1 h(k)
=
tan−1 f (n) − tan−1 f (1) + π
X
sign f (k),
where the sum is taken over all k between 1 and n for which f (k)f (k + 1) < −1.
Thus (2.2) is always correct up to an integral multiple of π. The restrictions on
the parameters in the examples described in this paper have the intent of keeping
f (k), k ∈ N of fixed sign.
Example 2.1. Let f (x) = ax + b, where a, b are such that f (x) ≥ 0 for x ≥ 1.
Then
a
h(x) =
(2.5)
a2 x2 + a(a + 2b)x + (1 + ab + b2 )
3
and (2.3) yields
(2.6)
∞
X
tan−1
k=1
a
a2 k 2 + a(a + 2b)k + (1 + ab + b2 )
=
π
− tan−1 (a + b).
2
Special cases: a = 1 and b = 0 give f (x) = x and h(x) = 1/(x2 + x + 1), and the
sum is
∞
X
π
1
(2.7)
=
.
tan−1 2
k +k+1
4
k=1
• a = 2 and b = 0 give f (x) = 2x and h(x) = 2/(2x + 1)2 , so that
∞
X
(2.8)
tan−1
k=0
2
(2k + 1)2
=
π
.
2
Differentiating (2.6) with respect to a yields
∞
X
pa,b (k)
(2.9)
k=1
qa,b (k)
=
1
,
1 + (a + b)2
where
= a2 k 2 + a2 k − (1 + b2 )
pa,b (k)
and
qa,b (k)
= a4 k 4 + 2a3 (a + 2b)k 3 + a2 (2 + a2 + 6ab + 6b2 )k 2 +
2a(a + 2b)(1 + ab + b2 )k + (1 + b2 )(1 + a2 + 2ab + b2 ).
Special cases: a = 1 and b = 0 give
∞
X
k2 + k − 1
4
k + 2k 3 + 3k 2 + 2k + 2
=
k=1
1
.
2
• a = 1/2 and b = 1/3 give
∞
X
k=1
81k 4
+
9k 2 + 9k − 40
+ 1269k 2 + 1932k + 2440
378k 3
=
1
.
61
• Let a = −2b and differentiate with respect to b to produce
∞
X
4b2 k 2 − (1 + b2 )
1
=
.
16b4 k 4 + 8b2 (1 − b2 )k 2 + (1 + b2 )2
2(1 + b2 )
k=1
Mathematica evaluates this last sum.
Example 2.2. This example considers the quadratic function f (x) = ax2 + bx + c
under the assumption that f (k), k ∈ N has fixed sign. For instance this happens if
b2 − 4ac ≤ 0. Define
1 + ac + bc + c2
ab + b2 + 2ac + 2bc
a2 + 3ab + b2 + 2ac
2a(a + b)
a0
a1
a2
a3
:=
:=
:=
:=
a4
:= a2
4
GEORGE BOROS AND VICTOR H. MOLL
Then
(2.10)
h(x)
and thus
∞
X
tan−1
k=1
2ax + a + b
a4 x4 + a3 x3 + a2 x2 + a1 x + a0
=
2ak + a + b
a4 k 4 + a3 k 3 + a2 k 2 + a1 k + a0
π
− tan−1 (a + b + c).
2
=
Special cases: b = −a, c = a/2 yields
∞
X
tan−1
k=1
8ak
4a2 k 4 + (a2 + 4)
=
π
a
− tan−1 .
2
2
The particular case a = 1, b = −1 and c = 1/2 gives
∞
X
(2.11)
tan−1
k=1
8k
4k 4 + 5
=
π
1
− tan−1 .
2
2
• b = −a, c = 0 yields the sum
∞
X
(2.12)
tan−1
k=1
a2 k 4
2ak
− a2 k 2 + 1
=
π
,
2
which is independent of a.
Additional examples can be given by telescoping twice (or even more).
Corollary 2.2. Let f and h be related by
h(x)
(2.13)
Then
n
X
tan−1 h(k)
=
=
f (x + 1) − f (x − 1)
.
1 + f (x + 1)f (x − 1)
tan−1 f (n + 1) − tan−1 f (1) + tan−1 f (n) − tan−1 f (0).
k=1
In particular,
∞
X
(2.14)
tan−1 h(k)
= 2 tan−1 f (∞) − tan−1 f (1) − tan−1 f (0).
k=1
Proof. The relation (2.13) shows that
tan−1 h(k)
=
tan−1 h(k)
=
tan−1 f (k + 1) − tan−1 f (k − 1).
Thus
n
X
k=1
=
=
n
X
−1
tan f (k + 1) − tan−1 f (k − 1)
k=1
n
n
X
−1
X
−1
tan f (k + 1) − tan−1 f (k) +
tan f (k) − tan−1 f (k − 1)
k=1
−1
tan
k=1
f (n + 1) − tan−1 f (1) + tan−1 f (n) − tan−1 f (0).
5
Example 2.3. The evaluation
∞
X
2
(2.15)
tan−1 2
k
3π
4
=
k=1
corresponds to f (k) = k so that h(k) = 2/k 2 . This is the problem proposed by
Anglesio [1].
Example 2.4. Take f (k) = −2/k 2 so that h(k) = 8k/(k 4 − 2k 2 + 5). It follows
that
∞
X
1
8k
= π − tan−1 .
tan−1 4
k − 2k 2 + 5
2
k=1
This sum is part b) of the Problem proposed in [1].
Example 2.5. Take f (k) = −a/(k 2 + 1). Then h(k) = 4ak/(k 4 + a2 + 4), so that
∞
X
4ak
a
(2.16)
tan−1 4
= tan−1 + tan−1 a.
k + a2 + 4
2
k=1
Special cases: a = 1 yields
∞
X
(2.17)
tan−1
k=1
•a=
√
4k
+5
k4
2 gives
∞
X
(2.18)
−1
tan
k=1
=
π
1
+ tan−1 .
4
2
√
4 2k
k4 + 6
π
.
2
=
• Differentiating (2.16) with respect to a gives
∞
X
4k(k 4 + 4 − a2 )
8
2
k + 2(a + 4)k 4 + 16a2 k 2 + (a4 + 8a2 + 16)
=
k=1
(a2
3(a2 + 2)
.
+ 1)(a2 + 4)
The special case a = 0 yields
∞
X
(2.19)
k=1
k
k4 + 4
=
3
8
and a = 2 gives
(2.20)
∞
X
k=1
k5
k 8 + 16k 4 + 64k 2 + 64
=
9
.
80
Mathematica evaluates the first sum but not the second.
The examples described above are rather artficial. The interesting question is
to find f (x) given the function h. In general it is not possible to find f in closed
form. In the case of Example 2.3, we need to solve the functional equation
(2.21)
2 [1 + f (x − 1)f (x + 1)] = x2 [f (x + 1) − f (x − 1)] .
A polynomial solution of (2.21) must have degree at most 2 and trying f (x) =
ax2 + bx + c yields f (x) = x.
6
GEORGE BOROS AND VICTOR H. MOLL
3. The method of zeros
A different technique for the evaluation of arctangent sums is based on the factorization of the product
(3.1)
pn
n
Y
:=
(ak + ibk )
k=1
with ak , bk ∈ R. The argument of pn is given by
Arg(pn )
n
X
=
tan−1
k=1
bk
.
ak
Example 3.1. Let
(3.2)
pn (z)
n
Y
=
(z − zk )
k=1
be a polynomial with real coefficients. Then
(3.3)
Arg(pn (z))
n
X
=
tan−1
k=1
x − xk
.
y − yk
n
The special case pn (z) = z − 1 has roots at zk = cos(2πk/n) + i sin(2πk/n), so we
obtain
n
X
x − cos(2πk/n)
Arg(z n − 1) =
tan−1
(3.4)
.
y − sin(2πk/n)
k=1
Example 3.2. The classical factorization
(3.5)
sin πz
= πz
∞ Y
1−
k=1
z2
k2
yields the evaluation
(3.6)
∞
X
tan−1
k=1
2xy
k 2 − x2 + y 2
= tan−1
y
tanh πy
− tan−1
.
x
tan πx
Special cases: x = y yields
∞
X
(3.7)
tan−1
k=1
2x2
k2
√
• x = y = 1/ 2 gives
(3.8)
∞
X
tan−1
k=1
1
k2
=
=
π
tanh πx
− tan−1
.
4
tan πx
√
π
tanh (π/ 2)
√ ,
− tan−1
4
tan(π/ 2)
which corresponds to (1.3).
• x = y = 1/2 yields
(3.9)
∞
X
k=1
tan−1
1
2k 2
=
π
.
4
7
• Differentiating (3.7) gives
∞
X
k2
(3.10)
4
k + 4x4
π sin 2πx − sinh 2πx
.
4x cos 2πx − cosh 2πx
=
k=1
In particular, x = 1 yields
∞
X
(3.11)
k=1
k2
+4
π
coth π.
4
=
k4
The identity (3.10) is comparable to Ramanujan’s evaluation
√
√
∞
X
k2
π
sinh πx 3 − 3 sin πx
√
√
(3.12)
=
k 4 + x2 k 2 + x4
2x 3 cosh πx 3 − cos πx
k=1
discussed in [3], Entry 4 of Chapter 14.
Glasser and Klamkin [10] present other examples of this technique.
4. A functional equation
The table of sums and integrals [11] contains a small number of examples of
finite sums that involve trigonometric functions of multiple angles. In Section 1.36
we find
n
X
x
x
(4.1)
22k sin4 k = 22n sin2 n − sin2 x
2
2
k=1
n
X
(4.2)
k=1
x
1
sec2 k
2k
2
2
cosec2 x −
=
1
x
cosec2 n
2n
2
2
and Section 1.37 consists entirely of the two sums
n
X
1
x
1
x
(4.3)
tan k =
cot n − 2 cot 2x
2k
2
2n
2
k=0
(4.4)
n
X
1
x
tan2 k
22k
2
=
k=0
22n+2 − 1
1
x
+ 4 cot2 2x − 2n cot2 n .
3 · 22n−1
2
2
In this section we present a systemtatic procedure to analyze these sums.
Theorem 4.1. Let
(4.5)
F (x) =
∞
X
f (x, k)
and
G(x) =
k=1
∞
X
(−1)k f (x, k).
k=1
Suppose f (x, 2k) = νf (λ(x), k) for some ν ∈ R and a function λ : R → R. Then
(4.6)
F (x)
=
n
(2ν) F (λ
(n)
(x)) −
n−1
X
(2ν)j G(λ(j) (x)).
j=0
Proof. Observe that
F (x) + G(x) = 2
∞
X
f (x, 2k) = 2ν
k=1
Repeat this argument to obtain the result.
∞
X
k=1
f (λ(x), k) = 2νF (λ(x)).
8
GEORGE BOROS AND VICTOR H. MOLL
Example 4.1. Let f (x, k) = 1/(x2 + k 2 ), so that ν = 1/4 and λ(x) = x/2. Since
F (x) =
∞
X
k=1
x2
1
πx coth πx − 1
=
2
+k
2x2
and
∞
X
(−1)k
πx csch πx − 1
G(x) =
=
,
x2 + k 2
2x2
k=1
(4.6) yields, upon letting n → ∞,
∞
X
(4.7)
j=0
x
− 2j
sinh 2−j x
1−
=
x
.
tanh x
Now replace x by ln t, differentiate with respect to t, and set t = e to produce
∞
X
2j − coth 2−j
2j sinh 2−j
j=0
(4.8)
1 + 4e2 − e4
.
1 − 2e2 + e4
=
Now go back to (4.7), replace x by ln t, differentiate with respect to t, set t = ae,
differentiate with respect to a, and set a = e to produce
(4.9)
∞
X
2 − 22j + csch2 2−j − sech2 2−j
22j sinh21−j
j=0
e12 − 17e8 − 17e4 + 1
.
e12 − 3e8 + 3e4 − 1
=
Corollary 4.2. Let
(4.10)
F (x) =
∞
x
X
f
k
and
G(x) =
k=1
∞
X
(−1)k f
k=1
x
k
.
Then, for any n ∈ N,
(4.11)
F (x)
= 2−n F (2n x) +
n
X
2−k G(2k x).
k=1
In particular, if F is bounded, then
(4.12)
F (x)
=
∞
X
2−k G(2k x).
k=1
Proof. The function f (x/k) satisfies the conditions of Theorem 4.1 with ν = 1 and
λ(x) = x/2. Thus
F (x) = 2n F (x/2n ) −
n−1
X
2j G(x/2j ).
j=1
n
Now replace x by x/2 to obtain (4.11). Finally let n → ∞ to obtain (4.12).
The key to the proof of Theorem 4.1 is the identity F (x) + G(x) = 2νF (λ(x)).
We now present an extension of this result.
9
Theorem 4.3. Let F, G be functions that satisfy
(4.13)
F (x)
= r1 F (m1 x) + r2 G(m2 x)
for some parameters r1 , r2 , m1 , m2 . Then
(4.14)
r2
n
X
r1k−1 G(mk−1
m2 x) = F (x) − r1n F (mn1 x).
1
k=1
Proof. Replace x by m1 x in (4.13) to produce
F (m1 x)
= r1 F (m21 x) + r2 G(m2 m1 x),
which, when combined with (4.13), gives
F (x)
= r12 F (m21 x) + r1 r2 G(m1 m2 x) + r2 G(m2 x).
Formula (4.14) follows by induction.
We now present two examples that illustrate Theorem 4.3. These sums appear
as entries in Ramanujan’s Notebooks.
Example 4.2. The identity
(4.15)
cot x =
1
x 1
x
cot − tan
2
2 2
2
shows that F (x) = cot x, G(x) = tan x satisfy (4.13) with r1 = 1/2, r2 = −1/2 and
m1 = m2 = 1/2. We conclude that
(4.16)
n
X
2−k tan
k=1
x
2k
=
1
x
cot n − cot x.
2n
2
This is (4.3). It also appears as Entry 24, p. 364, of Ramanujan’s Third Notebook
as described in Berndt [4], p. 396. Similarly, the identity
(4.17)
sin2 (2x) =
4 sin2 x − 4 sin4 x
yields (4.1). The reader is invited to produce proofs of (4.2) and (4.4) in the style
presented here.
Example 4.3. The identity
cot x =
(4.18)
cot
x
− csc x
2
satisfies (4.13) with F (x) = cot x, G(x) = csc x and parameters r1 = 1, r2 =
−1, m1 = 1/2, m2 = 1. We obtain
(4.19)
n
X
k=1
csc
x
2k−1
=
cot
x
− cot x.
2n
This appears in the proof of Entry 27 of Ramanujan’s Third Notebook in Berndt
[4], p. 398.
10
GEORGE BOROS AND VICTOR H. MOLL
Example 4.4. The application of Theorem 4.1, or Corollary 4.2, requires an analytic expression for F and G. A source of such expressions is Jolley [12]. Indeed,
entries 578 and 579 are
∞
X
tan−1
2x2
k2
=
(−1)k−1 tan−1
2x2
k2
= −
(4.20)
k=1
∞
X
(4.21)
k=1
π
tanh πx
− tan−1
4
tan πx
π
sinh πx
+ tan−1
.
4
sin πx
These results also appear in [5], page 314. Applying one step of Proposition 4.2 we
conclude that
tanh x
tanh 2x
sinh 2x
(4.22)
2 tan−1
= tan−1
+ tan−1
.
tan x
tan 2x
sin 2x
We also obtain
n
n
X
sinh 2k x
−1 tanh x
−n
−1 tanh 2 x
(4.23)
2−k tan−1
=
tan
,
−
2
tan
sin 2k x
tan x
tan 2n x
k=1
and by the boundedness of tan−1 x conclude that
∞
X
sinh 2k x
tanh x
(4.24)
= tan−1
.
2−k tan−1
k
sin 2 x
tan x
k=1
Differentiating (4.23) gives
2
n
X
cos 2k x sinh 2k x − cosh 2k x sin 2k x
k=1
cos 2k+1 x − cosh 2k+1 x
sin 2x − sinh 2x
sin 2n+1 x − sinh 2n+1 x
+
.
cos 2x − cosh 2x cos 2n+1 x − cosh 2n+1 x
Letting n → ∞ and using the identity
=−
(4.25)
cos 2k+1 x − cosh 2k+1 x = −2 sin2 2k x + sinh2 2k x
yields
∞
X
cosh 2k x sin 2k x − sinh 2k x cos 2k x
k=1
sin2 2k x + sinh2 2k x
=
sech2 x tan x − tanh x sec2 x
+ sign x.
tan2 x + tanh2 x
For example, x = π gives
(4.26)
∞
X
csch 2k π
=
coth π − 1.
k=1
5. Reduction to telescoping
The sum
(5.1)
S(f )
=
n
X
f (k, t),
k=1
depending on the parameter t, is said to telescope if the summand can be written
in the form
(5.2)
f (k, t) = f1 (k, t) − f1 (k + m, t)
11
for a function f1 and m ∈ Z fixed. In this section we discuss a method to determine
if the sum S telescopes.
We first consider the case of a function f (k, t) that is rational in k. Then the
question of telescoping is decided by examining the partial fraction decomposition
of f . For simplicity, we assume that the poles of f are simple and we omit the
parameter t.
Proposition 5.1. Let
f (k)
(5.3)
r
X
=
j=1
aj
k − kj
be the partial fraction decomposition of f . Then S(f ) telescopes if and only if f
can be written as
s X
bj
bj
(5.4)
f (k) =
−
k − kj
k − kj − mj
j=1
where mj ∈ Z.
We now observe that if (5.1) telescopes, then so do the sums
n
X
∂
f (k, t)
∂k
(5.5)
and
k=1
n
X
∂
f (k, t).
∂t
k=1
In particular, the question of telescoping of the arctangent sum
n
X
(5.6)
tan−1 R(k)
S(n) =
k=1
for a rational function R is reduced to that of the rational sum
n
X
1
∂R
(5.7)
S1 (n) =
,
1 + R2 (k) ∂k
k=1
by differentiating with respect to k. For instance, the sum in
rational function with poles at
p
p
p
1
1
1
(±1 − 1 − 4i/a), (±1 + 1 − 4i/a), (±1 − 1 + 4i/a),
2
2
2
and these can be paired as required in Proposition 5.1.
(2.12) produces a
p
1
(±1 + 1 + 4i/a),
2
Example 5.1. Consider the sum
∞ X
2k − 1
2k 2 − 2k + 1
2
−1
S =
k tan−1 4
−
(k
+
1)tan
.
k − 2k 3 + k 2 − 1
k 4 − 2k 3 + k 2 + 1
k=1
To evaluate S we introduce
2k 2 − 2k + 1
2k − 1
A(k) = k tan−1 4
− (k 2 + 1)tan−1 4
3
2
k − 2k + k − 1
k − 2k 3 + k 2 + 1
and observe that
d3
(5.8)
A(k) = B(k) − B(k − 1),
dk 3
where
6
8(3k 3 − 5k − 8) 64(k 3 − k − 1)
(5.9)
B(k) = − 4
−
+
.
k +1
(k 4 + 1)2
(k 4 + 1)3
12
GEORGE BOROS AND VICTOR H. MOLL
Let C(k) be the function obtained by integrating B(k) three times:
(5.10)
C(k)
= −(k 2 + k + 1)tan−1 k 2 + P2 (k)
where P2 (k) is a polynomial of degree 2.
Integrating backwards we obtain
N X
2k 2 − 2k + 1
2k − 1
−1
2
−1
k tan
− (k + 1) tan
k 4 − 2k 3 + k 2 − 1
k 4 − 2k 3 + k 2 + 1
k=2
= C(N ) − C(1)
= −(N 2 + N + 1)tan−1 N 2 + Q2 (N ),
where Q2 (N ) = P2 (N ) − C(1) is another polynomial of degree 2.√We have started
the sum at k = 2 because k 4 − 2k 3 + k 2 − 1 has a zero at k = 12 ( 5 + 1) ∼ 1.618.
The polynomial Q2 (N ) = aN 2 + bN + c can be determined from
aN 2 + bN + c =
(5.11)
N
X
A(k) + (N 2 + N + 1)tan−1 N 2
k=2
by evaluating (5.11) at three distinct values of N . The result is
π
(5.12)
Q2 (N ) =
2N 2 + 2N − 1 .
4
Thus
N X
k tan−1
k=2
2k 2 − 2k + 1
2k − 1
− (k 2 + 1) tan−1 4
4
k − 2k 3 + k 2 − 1
k − 2k 3 + k 2 + 1
= −(N 2 + N + 1)tan−1 N 2 +
so that
∞ X
k tan−1
k=1
π
(2N 2 + 2N − 1),
4
2k 2 − 2k + 1
2k − 1
3π
2
−1
− (k + 1) tan
=1−
.
4
3
2
4
3
2
k − 2k + k − 1
k − 2k + k + 1
2
Note. Naturally this idea applies to a more general class of sums. For instance,
sums of the form
(5.13)
L(n)
=
n
X
p(k) ln R(k)
k=1
and
(5.14)
A(n) =
n
X
p(k) tan−1 R(k),
k=1
for a polynomial p, can be reduced to a sum with rational summand by succesive
differentiation.
13
6. A dynamical system
In this section we describe a dynamical system that appears in the evaluation of
arctangent sums. Define
n
n
X
X
1
tan−1 .
tan−1 k and yn = tan
xn = tan
k
k=1
k=1
Then x1 = y1 = 1 and
xn =
nyn−1 + 1
xn−1 + n
and yn =
.
1 − nxn−1
n − yn−1
Proposition 6.1. Let n ∈ N. Then
(
−yn
(6.1)
xn =
1/yn
if n is even
if n is odd
that is
(6.2)
tan
n
X
tan−1 k
= −tan
k=1
n
X
tan−1
k=1
1
k
if n is even and
(6.3)
tan
n
X
tan−1 k
=
cotg
k=1
n
X
tan−1
k=1
1
k
if n is odd.
Proof. The recurrence formulas for xn and yn can be used to prove the result
directly. A pure trigonometric proof is presented next. If n is even then
tan
2m
X
k=1
tan−1 k + tan
2m
X
k=1
tan−1
1
k
=
tan
2m
X
tan−1 k + tan
k=1
=
tan
2m
X
π/2 − tan−1 k
k=1
−1
tan
k=1
=
2m
X
k + tan πm −
2m
X
−1
tan
k
!
k=1
0.
A similar argument holds for n odd.
This dynamical system suggests many interesting questions. We conclude by
proposing one of them: Observe that x3 = 0. Does this ever happen again?
References
[1] ANGLESIO, J.: Elementary problem 10292. Amer. Math. Monthly 100, 1993, 291. Solution
by several authors Amer. Math. Monthly, 103, 270-272.
[2] BERNDT, B.C.: Ramanujan’s Notebooks, Part I. Springer-Verlag, New York, 1985.
[3] BERNDT, B.C.: Ramanujan’s Notebooks, Part II. Springer-Verlag, New York, 1989.
[4] BERNDT, B.C.: Ramanujan’s Notebooks, Part IV. Springer-Verlag, New York, 1994.
[5] BROMWICH, T.J.: An introduction to the theory of infinite series. 1926.
[6] CHAPMAN, R.J.: Elementary problem 3375. Amer. Math. Monthly 97, 1990, 239-240.
[7] CHRYSTAL, G.: Algebra, Part II, 2nd. ed., A. and C. Black, London, 1922, p. 357.
[8] DARLING, D.: Telescoping series of arctangents. Amer. Math. Monthly 103, 1996, 270-272.
[9] GLAISHER, J.W.L.: A theorem in trigonometry, Quart. J. Math. 15, 1878, 151-157.
[10] GLASSER, M.L. - KLAMKIN, M.S.: On some inverse tangent summations. Fibonacci Quarterly, 14, 1976, 385-388.
14
GEORGE BOROS AND VICTOR H. MOLL
[11] GRADSHTEYN, I.S. - RYZHIK, I.M.: Table of Integrals, Series and Products. Sixth Edition,
ed. Alan Jeffrey, assoc. ed. D. Zwillinger. Academic Press, 2000.
[12] JOLLEY, L.B.W.: Summation of series, 1961
[13] LONEY, S.L.: Plane Trigonometry, Part II, Cambridge University Press, Cambridge, 1893,
p. 206.
[14] MILLER, A. - SRIVASTAVA, H.M.: On Glaisher’s infinite sums involving the inverse tangent function. Fibonacci Quart., 1992, 290-294.
[15] SARKAR, A.: The sum of arctangents of reciprocal squares. Amer. Math. Monthly, 98, 1990,
652-653.
[16] SCHAUMBERGER, N.: An old arctangent series reappears. The College Math. J. Problem
399. May 1990, vol. 21, 253-254.
Department of Mathematics, Xavier University, New Orleans, LA 70125
E-mail address: [email protected]
Department of Mathematics, Tulane University, New Orleans, LA 70118
E-mail address: [email protected]