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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

for the "hiking" mix, there are 1,000 pieces in the mix, containing 390.8 g of fat, and 165 g of protein. if there is the same amount of almonds as cashews, how many of each item is in the trail mix?
ADNAN Reply
linear speed of an object
Melissa Reply
an object is traveling around a circle with a radius of 13 meters .if in 20 seconds a central angle of 1/7 Radian is swept out what are the linear and angular speed of the object
Melissa
test
Matrix
how to find domain
Mohamed Reply
like this: (2)/(2-x) the aim is to see what will not be compatible with this rational expression. If x= 0 then the fraction is undefined since we cannot divide by zero. Therefore, the domain consist of all real numbers except 2.
Dan
define the term of domain
Moha
if a>0 then the graph is concave
Angel Reply
if a<0 then the graph is concave blank
Angel
what's a domain
Kamogelo Reply
The set of all values you can use as input into a function su h that the output each time will be defined, meaningful and real.
Spiro
how fast can i understand functions without much difficulty
Joe Reply
what is inequalities
Nathaniel
functions can be understood without a lot of difficulty. Observe the following: f(2) 2x - x 2(2)-2= 2 now observe this: (2,f(2)) ( 2, -2) 2(-x)+2 = -2 -4+2=-2
Dan
what is set?
Kelvin Reply
a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size
Divya Reply
I got 300 minutes. is it right?
Patience
no. should be about 150 minutes.
Jason
It should be 158.5 minutes.
Mr
ok, thanks
Patience
100•3=300 300=50•2^x 6=2^x x=log_2(6) =2.5849625 so, 300=50•2^2.5849625 and, so, the # of bacteria will double every (100•2.5849625) = 258.49625 minutes
Thomas
158.5 This number can be developed by using algebra and logarithms. Begin by moving log(2) to the right hand side of the equation like this: t/100 log(2)= log(3) step 1: divide each side by log(2) t/100=1.58496250072 step 2: multiply each side by 100 to isolate t. t=158.49
Dan
what is the importance knowing the graph of circular functions?
Arabella Reply
can get some help basic precalculus
ismail Reply
What do you need help with?
Andrew
how to convert general to standard form with not perfect trinomial
Camalia Reply
can get some help inverse function
ismail
Rectangle coordinate
Asma Reply
how to find for x
Jhon Reply
it depends on the equation
Robert
yeah, it does. why do we attempt to gain all of them one side or the other?
Melissa
how to find x: 12x = 144 notice how 12 is being multiplied by x. Therefore division is needed to isolate x and whatever we do to one side of the equation we must do to the other. That develops this: x= 144/12 divide 144 by 12 to get x. addition: 12+x= 14 subtract 12 by each side. x =2
Dan
whats a domain
mike Reply
The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
Spiro
Spiro; thanks for putting it out there like that, 😁
Melissa
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.
Churlene Reply
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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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