A number c is a limit point of the sequence {an} if there is a
subsequence {ank} of {an} converging to c.
Let S be the set of all the limit points of {an}. The limit infe-
rior, lim an, and the limit superior, lim an, of the sequence
{an} are defined as follows:
+oc if {an} is not bounded above,
—oo if {an} is bounded above and S = 0,
supS if {an} is bounded above and S ^ 0,
-oo if {an} is not bounded below,
-foo if {an} is bounded below and S = 0,
inf S if {an} is bounded below and S ^ 0.
lim an =
lim an
n—+oc
An infinite product Yl
an
is said to be convergent if there
7 1 = 1
exists no £ N such that an ^ 0 for n UQ and the sequence
{a
n o
a
n o +
i ... anQ+n} converges, as n oo, to a limit P0
other than zero. The number P = a ^ * an0-i ' Po is
called the value of the infinite product.
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