This document provides information about sequences and series in mathematics. It defines a sequence as an ordered set of real numbers denoted by a1, a2, a3, etc. where an represents the general term. Sequences can be finite or infinite. A sequence is said to converge to a limit l if the terms get arbitrarily close to l as n increases. Bounded sequences are those where there exists a number k such that an is always less than k. Monotonic sequences either always increase or decrease. A series is an infinite sum of terms of a sequence. The convergence, divergence or oscillation of a series depends on whether the partial sums sn tend to a finite limit, infinity, or fail to have a unique limit as

1. Mathematics
Presentation
Name: PRITAM MANDAL
Department: Computer Science & Engineering
Year: 2nd
Semester: 3rd
Roll no: 28100121048
Reg. no: 212810100110054

2. Sequences and
Series
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TOPIC:

3. SEQUENCE
S
(1) An ordered set of real numbers is called a sequence and is
denoted by
If the number of terms is unlimited , then the sequence is said to be an infinite
sequence and (an ) is its general term.
For instance (i) 1,3,5,7,…,(2n-1),…,
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a1,a2,a3,…...,an
(an )

4. (i) 1,1/2,1/3,…,1/n,...,
(ii) 1,-1,1,-1,…, (-1)(n-1) are infinite sequences.
(2) Limit. A sequence is said to tend to a limit l , if for every ε
>0 , a value N of n can be found such that
|an-l| < for n ≥ N We then write or simply as
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n→∞

5. (4) Bounded sequence: A sequence ( an ) is said to be bounded ,if there exists a
number k such that for every n.
(5) Monotonic sequence: the sequence ( an ) is said to increase steadily or to
decrease steadily according as or for all values of n .
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an+1≥an an+1≤an

6. “
Both increasing and decreasing sequences are called monotonic sequences.
A monotonic sequence always tends to a limit , finite or infinite.
Thus, a sequence which is monotonic and bounded is convergent.
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7. SERIE
S
(1) Definition : If be an infinite sequence of real
numbers , then Is called an infinite
series. An infinite series is denoted by and the sum of
its first n terms is denoted by .
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u1,u2,u3,…...un
u1+u2+u3+…...+un+.....
∞
sn

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(2) Convergence, Divergence and Oscillation of a series :
Consider the infinite series
And let the sum of the first n terms be
Clearly , is a function of n and as n increases indefinitely
three possibilities arise.
sn

9. (i) If tends to a finite limit as , the series is
said to be convergent.
(ii) If sn tends to as ,the series is said to be
divergent .
(iii) If sn does not tend to a unique limit as , then the
series is said to be oscillatory or non-convergent.
sn

10. For your patience.
Thank You!
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