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11.4 Series and their notations

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  • Use summation notation.
  • Use the formula for the sum of the first n terms of an arithmetic series.
  • Use the formula for the sum of the first n terms of a geometric series.
  • Use the formula for the sum of an infinite geometric series.
  • Solve annuity problems.

A couple decides to start a college fund for their daughter. They plan to invest $50 in the fund each month. The fund pays 6% annual interest, compounded monthly. How much money will they have saved when their daughter is ready to start college in 6 years? In this section, we will learn how to answer this question. To do so, we need to consider the amount of money invested and the amount of interest earned.

Using summation notation

To find the total amount of money in the college fund and the sum of the amounts deposited, we need to add the amounts deposited each month and the amounts earned monthly. The sum of the terms of a sequence is called a series    . Consider, for example, the following series.

3 + 7 + 11 + 15 + 19 + ...

The n th  partial sum of a series is the sum of a finite number of consecutive terms beginning with the first term. The notation   S n   represents the partial sum.

S 1 = 3 S 2 = 3 + 7 = 10 S 3 = 3 + 7 + 11 = 21 S 4 = 3 + 7 + 11 + 15 = 36

Summation notation is used to represent series. Summation notation is often known as sigma notation because it uses the Greek capital letter sigma , Σ , to represent the sum. Summation notation includes an explicit formula and specifies the first and last terms in the series. An explicit formula for each term of the series is given to the right of the sigma. A variable called the index of summation is written below the sigma. The index of summation is set equal to the lower limit of summation , which is the number used to generate the first term in the series. The number above the sigma, called the upper limit of summation , is the number used to generate the last term in a series.

Explanation of summation notion as described in the text.

If we interpret the given notation, we see that it asks us to find the sum of the terms in the series   a k = 2 k for k = 1 through k = 5.   We can begin by substituting the terms for k and listing out the terms of this series.

a 1 = 2 ( 1 ) = 2 a 2 = 2 ( 2 ) = 4 a 3 = 2 ( 3 ) = 6 a 4 = 2 ( 4 ) = 8 a 5 = 2 ( 5 ) = 10

We can find the sum of the series by adding the terms:

∑ k = 1 5 2 k = 2 + 4 + 6 + 8 + 10 = 30

Summation notation

The sum of the first n terms of a series can be expressed in summation notation    as follows:

∑ k = 1 n a k

This notation tells us to find the sum of a k from k = 1 to k = n .

k   is called the index of summation    , 1 is the lower limit of summation    , and n is the upper limit of summation    .

Does the lower limit of summation have to be 1?

No. The lower limit of summation can be any number, but 1 is frequently used. We will look at examples with lower limits of summation other than 1.

Given summation notation for a series, evaluate the value.

  1. Identify the lower limit of summation.
  2. Identify the upper limit of summation.
  3. Substitute each value of k from the lower limit to the upper limit into the formula.
  4. Add to find the sum.

Using summation notation

Evaluate ∑ k = 3 7 k 2 .

According to the notation, the lower limit of summation is 3 and the upper limit is 7. So we need to find the sum of k 2 from k = 3 to k = 7. We find the terms of the series by substituting k = 3 , 4 , 5 , 6 ,   and 7 into the function k 2 . We add the terms to find the sum.

∑ k = 3 7 k 2 = 3 2 + 4 2 + 5 2 + 6 2 + 7 2 = 9 + 16 + 25 + 36 + 49 = 135
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Questions & Answers

Three charges q_{1}=+3\mu C, q_{2}=+6\mu C and q_{3}=+8\mu C are located at (2,0)m (0,0)m and (0,3) coordinates respectively. Find the magnitude and direction acted upon q_{2} by the two other charges.Draw the correct graphical illustration of the problem above showing the direction of all forces.
Kate Reply
To solve this problem, we need to first find the net force acting on charge q_{2}. The magnitude of the force exerted by q_{1} on q_{2} is given by F=\frac{kq_{1}q_{2}}{r^{2}} where k is the Coulomb constant, q_{1} and q_{2} are the charges of the particles, and r is the distance between them.
Muhammed
What is the direction and net electric force on q_{1}= 5µC located at (0,4)r due to charges q_{2}=7mu located at (0,0)m and q_{3}=3\mu C located at (4,0)m?
Kate Reply
what is the change in momentum of a body?
Eunice Reply
what is a capacitor?
Raymond Reply
Capacitor is a separation of opposite charges using an insulator of very small dimension between them. Capacitor is used for allowing an AC (alternating current) to pass while a DC (direct current) is blocked.
Gautam
A motor travelling at 72km/m on sighting a stop sign applying the breaks such that under constant deaccelerate in the meters of 50 metres what is the magnitude of the accelerate
Maria Reply
please solve
Sharon
8m/s²
Aishat
What is Thermodynamics
Muordit
velocity can be 72 km/h in question. 72 km/h=20 m/s, v^2=2.a.x , 20^2=2.a.50, a=4 m/s^2.
Mehmet
A boat travels due east at a speed of 40meter per seconds across a river flowing due south at 30meter per seconds. what is the resultant speed of the boat
Saheed Reply
50 m/s due south east
Someone
which has a higher temperature, 1cup of boiling water or 1teapot of boiling water which can transfer more heat 1cup of boiling water or 1 teapot of boiling water explain your . answer
Ramon Reply
I believe temperature being an intensive property does not change for any amount of boiling water whereas heat being an extensive property changes with amount/size of the system.
Someone
Scratch that
Someone
temperature for any amount of water to boil at ntp is 100⁰C (it is a state function and and intensive property) and it depends both will give same amount of heat because the surface available for heat transfer is greater in case of the kettle as well as the heat stored in it but if you talk.....
Someone
about the amount of heat stored in the system then in that case since the mass of water in the kettle is greater so more energy is required to raise the temperature b/c more molecules of water are present in the kettle
Someone
definitely of physics
Haryormhidey Reply
how many start and codon
Esrael Reply
what is field
Felix Reply
physics, biology and chemistry this is my Field
ALIYU
field is a region of space under the influence of some physical properties
Collete
what is ogarnic chemistry
WISDOM Reply
determine the slope giving that 3y+ 2x-14=0
WISDOM
Another formula for Acceleration
Belty Reply
a=v/t. a=f/m a
IHUMA
innocent
Adah
pratica A on solution of hydro chloric acid,B is a solution containing 0.5000 mole ofsodium chlorid per dm³,put A in the burret and titrate 20.00 or 25.00cm³ portion of B using melting orange as the indicator. record the deside of your burret tabulate the burret reading and calculate the average volume of acid used?
Nassze Reply
how do lnternal energy measures
Esrael
Two bodies attract each other electrically. Do they both have to be charged? Answer the same question if the bodies repel one another.
JALLAH Reply
No. According to Isac Newtons law. this two bodies maybe you and the wall beside you. Attracting depends on the mass och each body and distance between them.
Dlovan
Are you really asking if two bodies have to be charged to be influenced by Coulombs Law?
Robert
like charges repel while unlike charges atttact
Raymond
What is specific heat capacity
Destiny Reply
Specific heat capacity is a measure of the amount of energy required to raise the temperature of a substance by one degree Celsius (or Kelvin). It is measured in Joules per kilogram per degree Celsius (J/kg°C).
AI-Robot
specific heat capacity is the amount of energy needed to raise the temperature of a substance by one degree Celsius or kelvin
ROKEEB
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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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