*This preview shows
page 1. Sign up
to
view the full content.*

**Unformatted text preview: **ar sequence {α1 , α2 , α3 , . . .} there is an associated sequence
of averages {µ1 , µ2 , µ3 , . . .} in which Y
P µ1 = α1 , µ2 = α1 + α2
,
2 ..., µn = α1 + α2 + · · · + αn
.
n
85 This sequence of averages is called the associated Ces`ro sequence, and when
a
a
limn→∞ µn = α, we say that {αn } is Ces`ro summable (or merely summable )
to α. It can be proven (Exercise 7.10.11) that if {αn } converges to α, then
{µn } converges to α, but not conversely. In other words, convergence implies
summability, but summability doesn’t insure convergence. To see that a sequence
can be summable without being convergent, notice that the oscillatory sequence
{0, 1, 0, 1, . . .} doesn’t converge, but it is Ces`ro summable to 1/2, which is the
a
mean value of {0, 1}. This is typical because averaging has a smoothing eﬀect
so that oscillations that prohibit convergence of the original sequence tend to be
smoothed away or averaged out in the Ces`ro sequence.
a O
C 85 Ernesto Ces`ro (1859–1906) was an Italian mathematician who worked mainly in diﬀerential
a
geometry but also contributed to number theory, divergent series, and mathematical physics.
After studying in Naples, Li`ge, and Paris, Ces`ro received his doctorate from the University
e
a
of Rome in 1887, and he went on to occupy the chair of mathematics at Palermo. Ces`ro’s
a
most important contribution is considered to be his 1890 book Lezione di geometria intrinseca ,
but, in large part, his name has been perpetuated because of its attachment to the concept of
Ces`ro summability.
a Copyright c 2000 SIAM Buy online from SIAM
http://www.ec-securehost.com/SIAM/ot71.html It is illegal to print, duplicate, or distribute this material
Please report violations to meyer@ncsu.edu Buy from AMAZON.com
7.10 Diﬀerence Equations, Limits, and Summability
http://www.amazon.com/exec/obidos/ASIN/0898714540 631 Similar statements hold for general sequences of vectors and matrices (Exercise 7.10.11), but Ces`ro summability is particularly interesting when it is
a
applied to the seq...

View
Full
Document