‫محاضرات التفاضل والتكامل المتقدم الصف الثاني‬
Infinite Sequence
Infinite Sequences
Definition
An infinite sequence is a function whose domain is the set of all positive integer numbers (
the natural numbers ) , i.e. the function a: N→ S is called a sequence and can be displayed in
the form
a(1), a(2), … , a(n), …
The value a(n) is called the nth term of the sequence and it is usually written as an. The whole
sequence is denoted by
< an > = < a1 , a2 , … , an , … >
or
{ an } = { a1 , a2 , … , an , … }
or
( an ) = ( a1 , a2 , … , an , … )
Remarks:
1. A sequence is an ordered list of objects (or events). Like a set, it contains members
(also called elements or terms).
2. A sequence is a discrete function.
3. A finite seq. with terms in a set S is a function from {1,2,3,…,n} to S for some n>0.
4. If S= R (the real numbers set) then the sequence is called a real seq. , and if S=Q (the
rational numbers set) then a sequence is called the rational seq. , and if S=Z (the integer
numbers set) then the sequence is called a integer seq. , and so on..
Examples:
1. 〈1〉 =
2. 〈𝑛〉 =
3. 〈𝑛 − 2〉 =
4. 〈−2𝑛〉 =
1
5. 〈 〉 =
𝑛
2𝑛
6. 〈
〉=
𝑛+1
1
7. 〈 𝑛 〉 =
3
8. 〈𝑛𝑒 𝑛 〉 =
9. 〈cos(𝑛𝜋)〉 =
Rem: If the sequence is simple enough one can look at the first few terms and guess the
general rule for computing the nth term, for example :
1. 〈1 , 1 , 2 , 6 , 24 , 120 , … 〉 =
2. 〈 1 , 2 , 6 , 24 , 120 , … 〉 =
3. 〈1 , 0.1 , 0.01 , 0.001 , 0.0001 , … 〉 =
4. 〈1 , −1 , 1 , −1 , 1 , −1 , … 〉 =
5. 〈1 , 𝑥 ,
𝑥2
2
,
𝑥3
6
,…〉 =
1
‫محاضرات التفاضل والتكامل المتقدم الصف الثاني‬
Infinite Sequence
Graphing the sequence
We can graph the sequence in the form of discrete function of a set of points.
Examples: graph the following sequences:
1
1) 〈 〉
2) < n >
3) < 2 >
4) < (-1)n+1 >
𝑛
Solve :
2
‫محاضرات التفاضل والتكامل المتقدم الصف الثاني‬
Infinite Sequence
Bounded & Unbounded Sequences
Definition
An infinite sequence < an > is said to be :1. Bounded above if there is a number A s.t, an  A  nN and A is called upper bound.
2. Bounded below if there is a number B s.t, an  B  nN and B is called lower bound.
3. Bounded if there is a number M s.t, |an|  M  nN and M is called a bound.
4. Unbounded if it is not bounded.
Examples:
1) < n-2>
1
5) 〈 2 〉
𝑛
Solve :
2) < -1n-1>
1
6) 〈 〉
𝑛
𝑛3
4) 〈
3) < -n >
𝑛4 +1000
𝑛
8) 〈
7) < ln(1/n) >
〉
𝑛+1
3
〉
‫محاضرات التفاضل والتكامل المتقدم الصف الثاني‬
Infinite Sequence
Increasing & Decreasing Sequences
Definition
An infinite sequence < an > is said to be :1. Increasing if anan+1  nN.
2. Decreasing if anan+1  nN.
3. Monotonic if it is increasing or decreasing.
Examples:
1) < n >
2) < -n >
3) < -1n+1 >
4) 〈
5) < cos(nπ) >
Solve :
6) < 1/n >
7) 〈
𝑛3
〉
𝑛+1
𝑛4 +10000
4
𝑛
〉
8) < ln(n) >