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Series

In this section we simply work on the concept of adding up the numbers belonging to arithmetic and geometric sequences. We call the sum of any sequence of numbers a series .

Some basics

If we add up the terms of a sequence, we obtain what is called a series . If we only sum a finite amount of terms, we get a finite series . We use the symbol S n to mean the sum of the first n terms of a sequence { a 1 ; a 2 ; a 3 ; ... ; a n } :

S n = a 1 + a 2 + a 3 + ... + a n

For example, if we have the following sequence of numbers

1 ; 4 ; 9 ; 25 ; 36 ; 49 ; ...

and we wish to find the sum of the first 4 terms, then we write

S 4 = 1 + 4 + 9 + 25 = 39

The above is an example of a finite series since we are only summing 4 terms.

If we sum infinitely many terms of a sequence, we get an infinite series :

S = a 1 + a 2 + a 3 + ...

Sigma notation

In this section we introduce a notation that will make our lives a little easier.

A sum may be written out using the summation symbol . This symbol is sigma , which is the capital letter “S” in the Greek alphabet. It indicates that you must sum the expression to the right of it:

i = m n a i = a m + a m + 1 + ... + a n - 1 + a n

where

  • i is the index of the sum;
  • m is the lower bound (or start index), shown below the summation symbol;
  • n is the upper bound (or end index), shown above the summation symbol;
  • a i are the terms of a sequence.

The index i is increased from m to n in steps of 1.

If we are summing from n = 1 (which implies summing from the first term in a sequence), then we can use either S n - or -notation since they mean the same thing:

S n = i = 1 n a i = a 1 + a 2 + ... + a n

For example, in the following sum,

i = 1 5 i

we have to add together all the terms in the sequence a i = i from i = 1 up until i = 5 :

i = 1 5 i = 1 + 2 + 3 + 4 + 5 = 15

Examples

  1. i = 1 6 2 i = 2 1 + 2 2 + 2 3 + 2 4 + 2 5 + 2 6 = 2 + 4 + 8 + 16 + 32 + 64 = 126
  2. i = 3 10 ( 3 x i ) = 3 x 3 + 3 x 4 + ... + 3 x 9 + 3 x 10
    for any value x .
Notice that in the second example we used three dots (...) to indicate that we had left out part of the sum. We do this to avoid writing out every term of a sum.

Some basic rules for sigma notation

  1. Given two sequences, a i and b i ,
    i = 1 n ( a i + b i ) = i = 1 n a i + i = 1 n b i
  2. For any constant c that is not dependent on the index i ,
    i = 1 n c · a i = c · a 1 + c · a 2 + c · a 3 + ... + c · a n = c ( a 1 + a 2 + a 3 + ... + a n ) = c i = 1 n a i

Exercises

  1. What is k = 1 4 2 ?
  2. Determine i = - 1 3 i .
  3. Expand k = 0 5 i .
  4. Calculate the value of a if:
    k = 1 3 a · 2 k - 1 = 28

Finite arithmetic series

Remember that an arithmetic sequence is a set of numbers, such that the difference between any term and the previous term is a constant number, d , called the constant difference :

a n = a 1 + d ( n - 1 )

where

  • n is the index of the sequence;
  • a n is the n t h -term of the sequence;
  • a 1 is the first term;
  • d is the common difference.

When we sum a finite number of terms in an arithmetic sequence, we get a finite arithmetic series .

The simplest arithmetic sequence is when a 1 = 1 and d = 0 in the general form [link] ; in other words all the terms in the sequence are 1:

a i = a 1 + d ( i - 1 ) = 1 + 0 · ( i - 1 ) = 1 { a i } = { 1 ; 1 ; 1 ; 1 ; 1 ; ... }

If we wish to sum this sequence from i = 1 to any positive integer n , we would write

i = 1 n a i = i = 1 n 1 = 1 + 1 + 1 + ... + 1 ( n times )

Since all the terms are equal to 1, it means that if we sum to n we will be adding n -number of 1's together, which is simply equal to n :

Questions & Answers

what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
How can I make nanorobot?
Lily
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
how can I make nanorobot?
Lily
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
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Source:  OpenStax, Siyavula textbooks: grade 12 maths. OpenStax CNX. Aug 03, 2011 Download for free at http://cnx.org/content/col11242/1.2
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