Convergence and Divergence of Infinite Series

Series:

The sum of an indicated number of terms in a sequence is called a series. Series is widely used in calculus and mathematical analysis. Series are also used in many fields like physics statistics finance etc.

History of series:

  • Archimedes  a Greek mathematician produce the first  known summation of an infinite Series with a method that is used in the field of calculus today.He used the method of exhaustion to find the area under the arc of a parabola with the summation of infinite series.
  • In 17th century the Mathematician James Gregory work in new decimal system om infinite series and published many Maclaurine series.
  • In 1715 the mathematician Brook Taylor gave a general method for constructing the Taylor series for all function for which they exist.

Example:

The sum of the first seven terms of the sequence {n²} is the series

1+4+9+16+25+36+49.

The above series is also named as the 7th partial sum of the sequence {n²}.

If the number of terms in a series is finite then the series is called a finite series while a series consisting of an unlimited number of terms is an infinite series.

Notation:

The Greek letter E is used to represent the sum of series. This notation is called summation or sigma. For example, the series 3+6+9+12+15 can be represented as 3\overset\infty{\underset{n=1}{\sum a_n}} where the n is called the index of summation.

Sum of first n terms of an Arithmetic series:

for any sequence {a_n} we have

S_n = a_1 + a_2 +a_3 +….+ a_n

if {an} is an A.P then S_n can be written with usual notation as

S_n = a_1 +( a_1 +d) + ( a_1 + 2d)+ …. +( a_n -2d) +( a_n -d) + a_n (1)

if we write the terms of series in the reverse order the sum of nterms remain the same that is

S_n = a_n +( a_n -d) = ( a_n -2d)+…….( a_1 +2d) = (a_1 +d) + a_1 (2)

adding (1) and (2) we get

2 S_n = ( a_1 + a_n ) +( a_1 + a_n ) +( a_1 + a_n ) =….=( a_1 + a_n ) +( a_1 + a_n ) +( a_1 + a_n )

S_n = ( a_1 + a_n ) +( a_1 + a_n )….to n terms

n( a_1 + a_n )

thus S_n = \frac n2 ( a_1 + a_n )

S_n =\frac n2[ a_1 + a_1 +(n-1)d]

S_n = \frac n2 [2 a_1 +(n-1)d]

Infinite Series:

Let {an} be a sequence .an expression of the form

a_1 + a_2 +a_3 +….+ a_n +….

containing infinitely many terms of the sequence of {an} is called an infinite series.

this series is symbolically written as.

\overset\infty{\underset{n=1}{\sum a_n}} or \overset\infty{\underset1{\sum a_n}}

here the a_n is called the nth term of the series.

Let S_n denote the sum of the first n terms of this series then we write

S_1 = a_1

S_2 = a_1 + a_2

S_3 = a_1 + a_2 + a_3

.

.

.

S_n = a_1 + a_2 + a_3 +……+ a_n

the sequence { S_n } is called the sequence of partial sums of the upper series and the number S_n is called the nth partial sum of the series \overset\infty{\underset1{\sum a_n}}

Converge and Diverge:

Let { S_n } be the sequence of partial sum of the series \overset\infty{\underset1{\sum a_n}} . if the sequence { S_n } converge to the limit s, then \overset\infty{\underset1{\sum a_n}} is said to be converge and S is called the sum of series .

if the sequence diverge then the series is said to be divergent.

To check the convergence of series we may have use some tests. These test are tell us about the convergence or divergence of series. These tests are following.

  • The Basic Comparision Test
  • The Limit Comparision Test
  • The Integral Test
  • The Ratio Test
  • The Cauchy Test

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