Page 34 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 2] SEQUENCES 25

LIMIT SUPERIOR, LIMIT INFERIOR

l
A number l is called the limit superior, greatest limit or upper limit (lim sup or lim) of the sequence

l
fu n g if infinitely many terms of the sequence are greater than l while only a finite number of terms are

l
greater than l þ , where is any positive number.
A number l is called the limit inferior, least limit or lower limit (lim inf or lim) of the sequence fu n g if
infintely many terms of the sequence are less than l þ while only a finite number of terms are less than
l , where is any positive number.
These correspond to least and greatest limiting points of general sets of numbers.
If infintely many terms of fu n g exceed any positive number M,we define lim sup fu n g¼1. If
infinitely many terms are less than M, where M is any positive number, we define lim inf fu n g¼ 1.
If lim u n ¼1,we define lim sup fu n g¼ lim inf fu n g¼1.
n!1
If lim u n ¼ 1,we define lim sup fu n g¼ lim inf fu n g¼ 1.
n!1
Although every bounded sequence is not necessarily convergent, it always has a finite lim sup and
lim inf.
A sequence fu n g converges if and only if lim sup u n ¼ lim inf u n is finite.

NESTED INTERVALS
Consider a set of intervals ½a n ; b n Š, n ¼ 1; 2; 3; ... ; where each interval is contained in the preceding
one and lim ða n b n Þ¼ 0. Such intervals are called nested intervals.
n!1
We can prove that to every set of nested intervals there corresponds one and only one real number.
This can be used to establish the Bolzano–Weierstrass theorem of Chapter 1. (See Problems 2.22 and
2.23.)

CAUCHY’S CONVERGENCE CRITERION
Cauchy’s convergence criterion states that a sequence fu n g converges if and only if for each > 0we
can find a number N such that ju p u q j < for all p; q > N. This criterion has the advantage that one
need not know the limit l in order to demonstrate convergence.

INFINITE SERIES
Let u 1 ; u 2 ; u 3 ; ... be a given sequence. Form a new sequence S 1 ; S 2 ; S 3 ; ... where

S 1 ¼ u 1 ; S 2 ¼ u 1 þ u 2 ; S 3 ¼ u 1 þ u 2 þ u 3 ; ... ; S n ¼ u 1 þ u 2 þ u 3 þ þ u n ; ...

where S n , called the nth partial sum,is the sum of the first n terms of the sequence fu n g.
The sequence S 1 ; S 2 ; S 3 ; ... is symbolized by

1
X
u n
u 1 þ u 2 þ u 3 þ ¼
n¼1
which is called an infinite series. If lim S n ¼ S exists, the series is called convergent and S is its sum,
otherwise the series is called divergent.
n!1
Further discussion of infinite series and other topics related to sequences is given in Chapter 11.