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1. Limit of the sequence
Definition 1
A sequence can be thought of as a list of numbers written in a
definite order y
1
,y
2
, ... y
n
,...
Definition 2
The number y
1
is called the first term, y
2
is the second term,
and in general y
n
is the nth term.
A sequence can also be defined as a function whose domain is the
set of all positive integers. We usually write y
n
(or x
n
) instead of the
function notation for the (real) value at n.
Notation 1
The sequence { y
1
,y
2
,...} is also denoted by {y
n
} or {y
n
}
∞
n=1
.
Definition 3
A sequence { x
n
} is called increasing if x
n+1
>x
n
for all n 1,
that is x
1
<x
2
<...<x
n
<.... It is called decreasing if x
n+1
<x
n
for all n.
Definition 4
Non-increasing means x
n+1
x
n
, non-decreasing: x
n+1
x
n
.
Definition 5
A sequence x
n
is called monotonic if it is either non-decreasing
or non-increasing. It is called strictly monotonic if it is either decreasing
or increasing.
Definition 6
A sequence {x
n
} is called bounded if there is such a number M
that |x
n
| <M for all n 1.
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