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limit of real number sequence∗
pahio†
2013-03-22 2:02:13
An endless real number sequence
a1 , a2 , a3 , . . .
(1)
has the real number L as its limit, if the distance between L and an can be
made smaller than an arbitrarily small positive number ε by chosing the ordinal
number n of an sufficiently great, i.e. greater than a number N (the size of
which depends on the value of ε); accordingly
|L − an | < ε when
n > N.
Then we may denote
lim an = L
n→∞
(2)
or equivalently
an → L as
n → ∞.
(3)
Remark 1. One should not think, that an = L when n = ∞. The symbol
“∞” represents no number, one cannot set it for the value of n. It’s only a
question of allowing n to exceed any necessary value.
Example 1. Using the notation (2) we can write a result
lim
n→∞
2n
= 2.
n+1
It’s a question of that the real number sequence
2 4 6
, , , ...
2 3 4
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1
1998
has the limit value 2 (e.g. the nine hundred ninety-ninth member 1000
= 1.998
is already “almost” 2!). For justificating the result, let ε be an arbitrary positive
number, as small as you want. Then
2 − 2n = 2n+2 − 2n = 2 = 2 < ε,
(4)
n+1
n+1
n+1
n+1 n+1
when n is chosen so big that
n>
2
−1.
ε
(5)
The condition (5) is obtained from (4) by solving this inequality for n. In this
case, we have N = 2ε −1.
Example 2. The so-called decimal expansions, i.e. endless decimal numbers, such as
3.14159265 . . . = π,
0.636363 . . . ,
0.99999 . . . ,
(6)
are, as a matter of fact, limits of certain real number sequences. E.g. the last
of these is related to the sequence
0.9, 0.99, 0.999, . . .
(7)
which may be also written as
1−
1
1
1
, 1− 2 , 1− 3 , . . .
10
10
10
The limit of (7) is 1. Actually, if ε > 0, the distance between 1 and the nth
member of (7) is
1 − 1− 1 = 1 < ε,
n
10 10n
when 10n > 1ε , i.e. when n > − log10 ε = N .
The endless decimal notations (6) and others are, in fact, limit notations —
no finite amount of decimals in them suffices to give their exact values.
Remark 2. In both of the above examples, no of the sequence members
was equal to the limit, but it does not need always to be so; thus for example
1+(−1)n
=0
n→∞
2n
lim
and every other member of the sequence in question is 0.
2
Infinite limits of real number sequences
There are sequences that have no limit at all, for example 1, −1, 1, −1, 1, −1, . . ..
Some real number sequences (1) have the property, that the member an may
exeed every beforehand given real number M if one takes n greater than some
value N (which depends on M ):
an > M
when
n > N.
Then we write
lim an = ∞.
n→∞
Similarly, the sequence (1) may be such that for each positive M there is N
such that
an < −M when n > N,
and then we write
lim an = −∞.
n→∞
E.g.
lim n2 = ∞,
n→∞
lim (1−n) = −∞.
n→∞
3