A Simple Proof of the Formula ∑∞k = 1 = π2/6
Author(s): Ioannis Papadimitriou
Source: The American Mathematical Monthly, Vol. 80, No. 4 (Apr., 1973), pp. 424-425
Published by: Mathematical Association of America
Stable URL: http://www.jstor.org/stable/2319092 .
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424
[April
IOANNIS PAPADIMITRIOU
Proof: Considertheidentity12(12 + 1) = (2,12) j2.
If 1212 is irrationalthenwe are finished.If not,then 1212 is rational. Hence
(1212)1V2 is irrational,and 12(/2 + 1) is the examplein thiscase.
Thereis also a simpleidentityby meansof whichit can be provedthata rational
numberraised to an irrationalpower may be irrational.But perhaps the reader
would enjoyfindingthisone himself.
References
Izv. Akad. Nauk SSSR, Ser. Mat.,
numbers,
1. R. Kuzmin,On a newclass of transcendental
7 (1930) 585-597.
Magazine,39( 1966) 111, 134.
2. Mathematics
3. ScriptaMathematica,19 (1953) 229.
00
A SIMPLE PROOF OF THE FORMULA
, k
k= 1
2 =
712/6
Athens, Greece
IOANNISPAPADIMITRIOU,
Start with the inequalitysinx <x <tan x for 0 < x < t/2, take reciprocals,
and square each memberto obtain
cot2x
<
1 + Cot2x.
1/x2 <
Now put x = ki/(2m+ 1) wherek and m are integers,1 ? k ? m, and sum on
k to obtain
(1)
+ 1)2
Xcot2k7-<(2m
k=
2m+1
m
1<m
kir
mot
k=k
2m+1
k-l
But since we have
E
(2)
t2
kt
m(2m-1)
_
(a proof of (2) is given below) relation(1) gives us
3
n2'
k=l
Multiplythis relationby 72/(4m2) and let m
m 1
lim
m-oo k=1
-
J
3
k2
oo to obtain
-
=6
r
6
Proof of (2). By equating imaginaryparts in the formula
cos nO+ i sin nO = (cos 0 + i sin O)n = sinno(cot0 + i)n
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1973]
425
CLASSROOM NOTES
sin0n
=
k=Ok
()
ikcotn ko,
we obtain the trigonometric
identity
sinnO = sin ((7)cotn'-
-
() cotn3 + (n)ctn-5o-+
Take n = 2m + 1 and writethisin the form
(3)
sin(2m+ 1)0 = =sin2m+1I OPm(cot2O)
with 0 < 0 < 2t
wherePmis the polynomialof degreem givenby
Pm(x)= (2
+ 1 )xm - (2
+ 1)xm- 1 + (2
+ 1)
xm-2-
+
Since sin0 # 0 for 0 < 0 < it/2, equation (3) shows thatPm(cot20)= 0 if and only
if (2m + 1)0 = kirfor some integerk. ThereforePm(x) vanishesat the m distinct
points Xk = cot2 rk/(2m
+ 1) for k = 1,2, ***,m. These are all the zeros of Pm(x)
and theirsum is
cot2
irk
_
2m + 1
(2m + 1
m(2m-1)
which proves (2).
to theMONTHLY
NOTE. Thispaperwas translated
froma Greekmanuscript
andcommunicated
on behalfof theauthorby Tom M. Apostol,California
of Technology.Afterthispaper
Institute
was written
it was learnedthatthe same proofwas discoveredindependently
and publishedin
Norwegianby Finn Holmein NordiskMatematisk
Tidskrift,
vol. 18 (1970),pp. 91-92. See also
A. M. Yaglom and I. M. Yaglom, Challenging
mathematical
problemswithelementary
solutions,
vol.II, Holden-Day,San Francisco,1967,problem145.
ANOTHER ELEMENTARY PROOF OF EULER'S FORMULA FOR C(2n)
TOM M. APOSTOL, California
ofTechnology
Institute
1. Introduction.
The classicformula
(1)
C(2n)=
E
10
-
=
)
)n
(27r)2nB2n
(2)
which expressesC(2n) as a rational multipleof 7r2n was discoveredby Euler [2].
The numbersBnare Bernoullinumbersand can be definedby the recursionformula
Bo=
1, Bn =
(
)Bs for n
2,
or equivalently,as the coefficients
in thepowerseriesexpansion
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