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1 + 1 + 1 + 1 +
In mathematics,
1 + 1 + 1 + 1 +
, also written , , or simply , is a divergentseries, meaning that its sequence of partial sumsdoes not converge to a limit in the real numbers.The sequence 1
n
can be thought of as ageometric series with the common ratio 1.Unlike other geometric series with rational ratio(except −1), it converges in neither the realnumbers nor in the
p
-adic numbers for some
p
.In the context of the extended real number linesince its sequence of partial sums increases monotonically without bound.Where the sum of
n
0
occurs in physical applications, it may sometimes beinterpreted by zeta function regularization, as the value at
s
= 0
of the Riemann zetafunctionThe two formulas given above are not valid at zero however, so one might try theanalytic continuationof the Riemann zeta function,
Using this one gets (given that
Γ(1) = 1
),where the power series expansion for
ζ
(
s
)
about
s
= 1
follows because
ζ
(
s
)
has a simple pole of residue one there. In this sense
1 + 1 + 1 + 1 +
⋯
=
ζ
(0) = −
12.Emilio Elizalde presents an anecdote on attitudes toward the series:In a short period of less than a year, two distinguished physicists, A. Slavnov and F. Yndurain, gave seminars inBarcelona, about different subjects. It was remarkable that, in both presentations, at some point the speaker addressedthe audience with these words: 'As everybody knows, 1 + 1 + 1 +
⋯
= −
12
.' Implying maybe:
If you do not know this,it is no use to continue listening.
[2]
The series 1 + 1 + 1 + 1 +
⋯
After smoothingAsymptotic behavior of thesmoothing. The
y
-intercept of the lineis −
12
.
[1]
1 − 1 + 2 − 6 + 24 − 120 + · · ·Harmonic series1. Tao, Terence (April 10, 2010),
The Euler-Maclaurin formula, Bernoulli numbers, the zeta function, and real-variableanalytic continuation
(http://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/), retrieved January 30, 20142. Elizalde, Emilio (2004). Cosmology: Techniques and Applications .
Proceedings of the II International Conferenceon Fundamental Interactions
. arXiv:gr-qc/0409076 (https://arxiv.org/abs/gr-qc/0409076).
OEIS sequence A000012 (The simplest sequence of positive numbers: the all 1's sequence)Retrieved from https://en.wikipedia.org/w/index.php?title=1_%2B_1_%2B_1_%2B_1_%2B_
⋯
&oldid=832701315
This page was last edited on 27 March 2018, at 14:40.
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See alsoNotesExternal links

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