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 a1, a2, a3, …, ai, … .
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 first n terms of a sequence is represented by
where i is called the index of summation, n is the upper limit of summation, and 1 is the lower limit of summation.
Given a sequence a1, a2, a3, a4, … .
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 ak from 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.
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). ∑nk=1 C=nC.
(i). Formula 2. ∑nk=1 k=½n(n+1).
(ii). Formula 3. ∑nk=1 k2 =⅙n(n+1)(2n+1).
(iii). Formula 4. ∑nk=1 k3 =[½n(n+1)]2
Important results on the sum of special sequences
(i) Sum of the first n natural numbers:
Σn=1+2+3+⋯+n=½n(n+1)
(ii) Sum of the squares of first n natural numbers.
Σn2=12+22+32+⋯+n2=⅙n(n+1)(2n+1)
(iii) Sum of cubes of first n natural numbers:
Σn3=13+23+33+⋯+n3=[½n(n+1)]2
Now that we have this list, let’s use them to compute cubic series.
Another Type of Sequence
For a sequence xi(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 a1, a2, a3, a4, … and b1, b2, b3, b4, … be sequences. Then for every positive integer n and a constant c, the following properties hold.
Properties of Summation Notation
Suppose {an} and {bn} are sequences so that the following sums are defined.
• ∑pn=m (an±bn ) =∑pn=m an ±∑pn=m bn
• ∑pn=m c⋅an=c∑pn=m an , for any real number c.
• ∑pn=m an =∑jn=m an +∑pn=j+1 an , for any natural number m≤j<j+1≤p.
• ∑pn=m an =∑p+rn=m+r a(n-r) , for any whole number r.
