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.

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.