Gandhinagar Institute of Technology

CalculasActive Learning Assignment

Sequnces & Series

Mechanical branchLG1

Group members:

Vrushang Sangani 140120119193

Chintan Sathvara

140120119195

Guided By: Prof.Bhumika maheshwari

Content

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

Sequences

• Sequences are ordered lists generated by a function, for example f(n) = 100n

(1), (2), (3),...

100,200,300,...

f f f

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

Sequences

A sequence is often specified by giving a formula forthe general term or nth term, an.

Example Find the first four terms for the sequence

Solution

1

2n

na

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

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, …

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.

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.

Series

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

Finite Series Infinite Series2+4+6+8+10 2+4+6+8+10+…

Convergent and Divergent Series

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 31

...n

n n ii

S a a a a a

1 2 31

... ...n ii

S a a a a a

nth term test for divergence:

Refernces

• www.google.co.in• http://en.wikipedia.org/• http://17calculus.com/infinite-series/nthterm

/#var000• http://www.dummies.com/how-to/content/h

ow-to-use-the-nth-term-test-to-determine-whether-.html

• http://www.mathsisfun.com/algebra/sequences-series.html

Thank You