This document discusses sequences and series in calculus. It defines sequences as ordered lists generated by a function with domain of natural numbers. Sequences can be finite or infinite. A sequence converges if its terms get closer to a number, and diverges otherwise. A series is the sum of terms in a sequence. If the number of terms is finite, it is a finite series, and infinite otherwise. A series converges if the sum converges, and diverges otherwise. The document introduces tests for determining convergence of sequences and series, including the nth term test for divergence. It provides examples to illustrate these concepts.

1. Calculas
Active Learning Assignment
Sequnces & Series
Mechanical branch
LG1
Group members:
Vrushang Sangani 140120119193
Chintan Sathvara 140120119195

2. Content
• Infinite Sequences and Series
• Introduction of Convergence, Divergence of
Sequences and Infinite Series
• The nth term test for Divergence

3. Sequences
• Sequences are ordered lists generated by a
function, for example f(n) = 100n
(1), (2), (3),...
100,200,300,...
f f f

4. Sequences
• A sequence is a function that has a set of natural
numbers as its domain.
• f (x) notation is not used for sequences.
• Sequences are written as ordered lists
• a1 is the first element, a2 the second element, and
so on
( )na f n
1 2 3, , , ...a a a

5. Sequences
A sequence is often specified by giving a formula for
the general term or nth term, an.
Example Find the first four terms for the sequence
Solution
1
2
n
n
a
n



1 2(1 1)/(1 2) 2/3, (2 1)/(2 2) 3/ 4a a       
3 4(3 1) /(3 2) 4/5, (4 1) /(4 2) 5/ 6a a       

6. Sequences
• A finite sequence has domain the finite set
{1, 2, 3, …, n} for some natural number n.
Example 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
• An infinite sequence has domain
{1, 2, 3, …}, the set of all natural numbers.
Example 1, 2, 4, 8, 16, 32, …

7. Convergent and Divergent
Sequences
• A convergent sequence is one whose terms get
closer and closer to a some real number. The
sequence is said to converge to that number.
• A sequence that is not convergent is said to be
divergent.

8. Series
• The sum of the terms in a sequence is called series.
• If the number of terms is finite then the series is
called finite series.
• If the number of terms is unlimited then the series is
called infinite series.

9. Series
• For Example:
Finite Seq. Infinite Seq.
2,4,6,8,10 2,4,6,8,10,…
Finite Series Infinite Series
2+4+6+8+10 2+4+6+8+10+…

10. Convergent and Divergent Series

11. Series and Summation Notation
• A finite series is an expression of the form
• and an infinite series is an expression of the
form
1 2 3
1
...
n
n n i
i
S a a a a a

      
1 2 3
1
... ...n i
i
S a a a a a



       

12. nth term test for divergence:

14. Refernces
• www.google.co.in
• http://en.wikipedia.org/
• http://17calculus.com/infinite-
series/nthterm/#var000
• http://www.dummies.com/how-
to/content/how-to-use-the-nth-term-test-to-
determine-whether-.html
• http://www.mathsisfun.com/algebra/sequenc
es-series.html

15. Thank
You