**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 17
^{th}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 7^{th} 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**