**Sigma Notation**

**Definition of Series**

A series is the sum of the terms of a finite or infinite sequence in a long form.

Consider the infinite sequence *a*_{1}, *a*_{2}, *a*_{3}, …, *a _{i}*, … .

1. The sum of the first *n* terms of the sequence is called a **finite** series or the ** nth partial sum** of the sequence and is denoted by

2. The sum of all the terms of the infinite sequence is called an **infinite series** and is denoted by

A convenient notation for the sum of the terms of a finite sequence is called **summation notation** or **sigma notation**. It involves the use of the uppercase Greek letter sigma, written as Σ.

Definition of Summation Notation

The sum of the firstnterms of a sequence is represented by

whereiis called theindex of summation,nis theupper limit of summation, and 1 is thelower limit of summation.

Given a sequence *a*_{1}, *a*_{2}, *a*_{3}, *a*_{4}, … .

We can write the sum of the first *n* terms using summation notation, or sigma notation. This notation derives its name from the Greek letter Σ (capital sigma, corresponding to our *S* for “sum”). Sigma notation is used as follows:

The left side of this expression is read, “The sum of

*a*from

_{k}*k*=1 to

*k=n*.”

The letter

*k*is called the

**index of summation**, or the

**summation variable**, and the idea is to replace

*k*in the expression after the sigma by the integers 1, 2, 3, … ,

*n*. and add the resulting expressions, arriving at the right side of the equation.

**Calculating With Sigma Notation**

We want to use sigma notation to simplify our calculations. To do that, we will need to know some basic sums. First, lets talk about the sum of a constant. (Notice here, that our upper limit of summation is *n*. *n* is not the index variable, here, but the highest value that the index variable will take.)

This is a sum of

*n*terms, each of them having a value

*C*. That is, We are adding

*n*copies of

*C*. This sum is just

*nC*. The other basic sums that We need are much more complicated to derive. Rather than explaining where they come from, we’ll just give you a list of the final formulas, that you should remember.

**Frequendly Used Formulae**:

(0). ∑

^{n}

_{k=1}

*C=nC*.

(i). Formula 2. ∑

^{n}

_{k=1}

*k*=½

*n*(

*n*+1).

(ii). Formula 3. ∑

^{n}

_{k=1}

*k*

^{2}=⅙

*n*(

*n*+1)(2

*n*+1).

(iii). Formula 4. ∑

^{n}

_{k=1}

*k*

^{3}=[½

*n*(

*n*+1)]

^{2}

**Important results on the sum of special sequences**

(i) Sum of the first *n* natural numbers:

Σ*n*=1+2+3+⋯+*n*=½*n*(*n*+1)

(ii) Sum of the squares of first *n* natural numbers.

Σ*n*^{2}=1^{2}+2^{2}+3^{2}+⋯+*n*^{2}=⅙*n*(*n*+1)(2*n*+1)

(iii) Sum of cubes of first *n* natural numbers:

Σ*n*^{3}=1^{3}+2^{3}+3^{3}+⋯+*n*^{3}=[½*n*(*n*+1)]^{2}

Now that we have this list, let’s use them to compute cubic series.

**Another Type of Sequence**

For a sequence *x _{i}*(

*i*=1, 2, …) where

*a*and

*b*are constants:

Which can be further generalised for sequences beginning at

*m*:

The following properties of sums are natural consequences of properties of the real numbers

**Properties Of Sums**

Let *a*_{1}, *a*_{2}, *a*_{3}, *a*_{4}, … and *b*_{1}, *b*_{2}, *b*_{3}, *b*_{4}, … be sequences. Then for every positive integer *n* and a constant *c*, the following properties hold.

**Properties of Summation Notation**

Suppose {*a _{n}*} and {

*b*} are sequences so that the following sums are defined.

_{n}• ∑^{p}_{n=m} (*a _{n}*±

*b*) =∑

_{n}^{p}

_{n=m}

*a*±∑

_{n}^{p}

_{n=m}

*b*

_{n}• ∑^{p}_{n=m} c⋅*a _{n}*=c∑

^{p}

_{n=m}

*a*, for any real number

_{n}*c*.

• ∑^{p}_{n=m} *a _{n}* =∑

^{j}

_{n=m}

*a*+∑

_{n}^{p}

_{n=j+1}

*a*, for any natural number

_{n}*m*≤

*j*<

*j*+1≤

*p*.

• ∑^{p}_{n=m} *a _{n}* =∑

^{p+r}

_{n=m+r}

*a*

_{(n-r)}, for any whole number

*r*.