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To make the above even more precise: L is the limit of x
n
,iffor
an arbitrary positive ε there exists such N (which might depend on
ε)that|x
n
− L| <εas soon as n N.
1.1. Limit Laws for Convergent Sequences
If {a
n
} and {b
n
} are convergent sequences and c is a constant,
then
lim
n→∞
(a
n
+ b
n
) = lim
n→∞
a
n
+ lim
n→∞
b
n
;
lim
n→∞
(a
n
− b
n
) = lim
n→∞
a
n
− lim
n→∞
b
n
;
lim
n→∞
c · a
n
= c lim
n→∞
a
n
;
lim
n→∞
(a
n
b
n
) = lim
n→∞
a
n
· lim
n→∞
b
n
;
lim
n→∞
a
n
b
n
=
lim
n→∞
a
n
lim
n→∞
b
n
, if lim
n→∞
b
n
=0,
lim
n→∞
c = c.
Let us now consider a few situations which involve unbounded
sequences.
Example 2
1) x
n
=1/
√
n, y
n
= n, x
n
· y
n
=
√
n →∞;
2) x
n
= a/n, y
n
= n, x
n
· y
n
= a → a;
3) x
n
=1/n
2
, y
n
= n, x
n
· y
n
=1/n → 0;
4) x
n
=(−1)
n
/n, y
n
= n, x
n
· y
n
=(−1)
n
, no limit.
In each case, we were in a 0 ·∞indeterminate situation.
Afewtechnicalities (when working with sequences).
Let x
n
=
n
2
+1
n
2
− 1
. When trying to apply the respective limit law,
we get an indeterminate
∞
∞
expression. A way to go is to divide both
5
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