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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

suppose you are the manager of a perfect competitive firm. your total cost of production is given by TC=100+Q2 suppose you are told that the price of their services is $60 1. how much output should you produce to maximise profit? show calculations 2. determine the level of profit
Niza Reply
220
ezhilarasan
what is price control
Raphael Reply
importance of scale of preference
Offei Reply
what are the factors that affect international organization
Imeobong Reply
Organizational structure,communication,mission
Ogunsola
what is international organization
Imeobong Reply
discuss how international organization affect business in Nigeria
Imeobong
An international organization is an organization with an international membership, scope, or presence.
Avishek
thanks
Imeobong
importance of scale of preference
Offei
demand curve
Raphael
demand uuu
Raphael
uu
Raphael
supply curve
Raphael
what is price control
Raphael
what are the factors that affect demand and commodity
Beatrice Reply
if a firm stays on the same isoquant it means ? a. output missteps decrease b. output must stay. the same c. output must increase d. quantity of labour and capital employed must remain the same e. none
Niza Reply
A
Were
A
alyssa
B
SEMAN
A
Darren
B
Bertilla
D
Romeo
which one is the right answer pls
Bertilla
😂😂😂😂
Niza
it's bad
Niza
B
Foley
d
Fasae
b
Alicia
a
Mohammed
hi
DINA
d
DINA
d) none. because isoquants meant different combination of inputs give same level of out and higher the isoquants gives higher the level of output
ezhilarasan
okay
Bertilla
answ. d.quantity of labour and capital employed must remain equal
mohamed
d
Ayoka
B
jephter
analyse this table showing how this affect the law of diminishing demand return
michael Reply
What are diminishing marginal returns as they relate to costs?
michael
what are diminishing demand
Bertilla
please can you explain it to me because am new to economics
michael
the law of diminishing state that as more and more variable factor is added to a fixed cost, it will increase first and later its begins to fall until its get to zero. that were ur mc equal to zero
Fasae
thanks dr
Bertilla
The law of diminishing demand states that, if the price of a product is raised, a smaller quantity will be demanded and if the price of a product is lowered, a greater quantity will be demanded.
Were
The law of diminishing returns, also referred to as the law of diminishing marginal returns, states that in a production process, as one input variable is increased, there will be a point at which the marginal per unit output will start to decrease, ceteris puribus
Were
guys I need to know what is meant by basic economic mathematics and explain please
Lovemore Reply
It's also the usage of mathematical tools to express and analyse economic problems.
Rasaq
so basically it's the use of mathematical formulas to solve the problems under demand and supply right...
Lovemore
Yes, though economic analysis majorly has to deal with demand and supply forces in fact the back bone of economics rest on demand and supply but there are other functions like consumption function, implicit ,explicit, monotonic( constant, increasing, decreasing), transcendential, homogeneous, polyn
Rasaq
wat improvement do economic has with our life
Aminata Reply
it help us how to use our limited resources to satisfy our unlimited wants
Pagnol
Economics assit in understanding the pre-requisite for Growth and developmen and how the workings of the economy operates.
Rasaq
Economic helps us to manager our limited resources, how to expand it, etc over unlimited want
Eagle
Economics is the social Science in which we study how scarce resources are allocated for the betterment of human being.
Ali Reply
what is equilibrium point
Happiness Reply
where price and quantity demanded meet
Pagnol
is the point where price an quantity demanded meet
Aminata
why do consumer demand curve tend to move downward slope
Aminata
Equilibrium point is the point at which quantity demand equals quantity supply.
Rasaq
What is monopoly
Ezekiel Reply
Is a strict example of an imperfect market where there is only one seller and many buyer for a particular product.
Rasaq
what is economics
BUKENYA Reply
economics us behavior. Of people, nations, markets etc. It has much to do with reactions. watch MSNBC. Bloomberg. everyday they talk about how the markets react to an piece of news, legislation, interest rates etc. Interesting volatile stuff
TOM
but, it also react with science
ezhilarasan
economic is a science with study's the behavior of people,market and price
Bertilla
it's a science which study the behavior of people ,market n price
Bertilla
What is opportunity cost?
Junior
opp. cost~a benefit, profit or value of something that must be given up to acquire or achieve something else
Kim
or something that you foregone
Kim
who is the father of economic
Yirtutey Reply
Adam smith
shaikh
modern father is Adam smith
Bilal
Adam smith
Nazifi
Jeannette has $5 and $10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
August Reply
What is the expressiin for seven less than four times the number of nickels
Leonardo Reply
How do i figure this problem out.
how do you translate this in Algebraic Expressions
linda Reply
why surface tension is zero at critical temperature
Shanjida
I think if critical temperature denote high temperature then a liquid stats boils that time the water stats to evaporate so some moles of h2o to up and due to high temp the bonding break they have low density so it can be a reason
s.
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
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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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