Significant insignificant digits!
Citation for published version (APA):
Post, K. A. (1982). Significant insignificant digits! (Eindhoven University of Technology : Dept of Mathematics : memorandum; Vol. 8216). Technische Hogeschool Eindhoven.
Document status and date: Published: 01/01/1982 Document Version:
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EINDHOVEN UNIVERSITY OF TECHNOLOGY
. Department of Mathematics and Computing Science
Memorandum 1982-16 October 1982
SIGNIFICANT INSIGNIFICANT DIGITS!
by
K.A. Post
Eindhoven University of Technology Department of Mathematics
and Computing Science
P.O. Box 513, 5600 MB Eindhoven the Netherlands
Significant insignificant digits
Recently Martin R. Powell [lJ observed a remarkable agreement between
cer-tain significant digits of ~ and of certain partial sums of its series
expan-sion
When S is defined to be the sum of the first n terms of this series Powell n
found
correct to 1 significant figure:
'IT = 3 } difference -1 correct to 4 s.f.: ~ = 3.142 S10
=
3.042}
-) .. difference 10 correct to 7 s.f.: 'IT = 3.141593 S == 3.l31593 102 } difference 10-2 correct to 16 s.f.: ~ 3.141592653589793 S == 3.141582653589793 105 } difference 10-5 •2
-This last result took 13! hours on the school's computer, so Powell asked whether this phenomenon is systematic or pure accident.
In this note we shall prove that Powell's observations can be extended:
Clearly one has
I
I
4 dx 11"=
2 0 + x and I n-II
(_I)k 2k S=
'L
• 4x dx, n k=O 0 so that 1 4x2n (1) S (-I) nJ
dx • 11" -=
n 1 + x 2 0We shall derive upper and lower bounds for the last integral: For x ~ 0
2n 2n-) 2 • 2n-1 2
one has 4x ~ 2x (l+x ), S1nce 2x (I-x) ~
o.
Hence we obtain
1
(2)
J
no
The difference D of the two integrals in (2) satisfies:
o
< D 1 2J
o
2n-I 2 x (1-2x+x ) dx < 2 2 I + x If
o
=l\_1 __
2_ +~)
2n 2n+l 2n+2 1 1 ~--~~--~ < --- • n(n+I) (2n+l) 2n3Therefore 1 1 - - - - < n 2n3 I I - - - - < n 2n3 1
J
o
3 -4x2n d<.!..
2 x n + x n I (-I) (TI-S ) < - • n n or, by (l)This extends Powell's results for all nEE.
[IJ Martin R. Powell. Significant insignificant digits? Math. Gazette October 1982, p. 220-221.