Sigma Notation: Learn How to Evaluate Expressions

In summary: In this case, the pattern is that the coefficient alternates between -1 and 1. So you can write it like this: 3^3 - 3^4 + 3^5 - ... - 3^100
  • #1
missadorkable
3
0
Express this in sigma notation?
3^3 - 3^4 + 3^5 - ... - 3^100

Evaluate these two sigmas?
n
∑ (i-2)^2
i =1

n
∑ (4-i^2)
i =1

I don't really understand sigma notation so I'm really interested in the process and explanation of how to do it. Any help would be greatly appreciated!
 
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  • #2
Did your course materials cover how to add up the squares of consecutive integers?
Do they give a formula for
[tex] \sum_{i=1}^n i^2 [/tex]
 
  • #3
yes.

n
∑ (i^2) = n(n+1)/2
i =1

How do I apply this? :S
 
  • #4
For example

[tex] \sum_{i=1}^n 3i^2 + 2i + 1 [/tex]

The summation can be distributed to obtain

[tex] = 3 \sum_{i=1}^n i^2 + 2 \sum_{i=1}^n i + \sum_{i=1}^n 1 [/tex]

[tex] = 3 ( n (n+1)/2) + 2 \sum_{i=1}^n i + n [/tex]

The materials probably give the the formula for [tex] \sum_{i=1}^n i [/tex] and you can use it also.

In the problems you asked about, you need to multiply out the expressions like [tex] (i-2)^2 [/tex] before you distribute the summation.
 
  • #5
Alright, I think I understand what you're saying. What about the first question with expressing 3^3 - 3^4 + 3^5 - ... - 3^100 as sigma notation? Is there a formula that could be used to acquire the expression for the sigma notation?
 
  • #6
missadorkable said:
Is there a formula that could be used to acquire the expression for the sigma notation?

No, there isn't a mechanical procedure for determining the formula (the function of i) that would be used in the sigma notation to get that series. You have to do it by trial and error. The terms alternate, so it could involve a negative number raised to a power. You can throw in a factor of [tex] (-1)^i [/tex] or [tex] (-1)^{i+1} [/tex] in the formula if it is needed to make the signs come out correctly. The things that are increasing by 1 in each term should tell you that the formula involves [tex]i [/tex] as an exponent. It looks like the first exponent starts as 3 not as 1. If your materials want you to start all summations with [tex] i = 1 [/tex] then use the expression [tex] i+2 [/tex] in the exponent to make the first term have an exponent of 3.
 
  • #7
missadorkable said:
yes.

n
∑ (i^2) = n(n+1)/2
i =1

How do I apply this? :S

This is wrong. n(n+1)/2 is the sum of the first n integers.

To write the first in sigma notation, just look at the pattern. you've got a coefficient that is oscliating between -1 and 1, you have some odd numbers in the denominator. Just try to find a pattern.
 

What is sigma notation?

Sigma notation is a mathematical notation used to represent the sum of a series of numbers. It is represented by the Greek letter sigma (Σ) followed by an expression that indicates the starting value of the series, the ending value, and the pattern for the terms in the series.

How do you evaluate expressions using sigma notation?

To evaluate an expression using sigma notation, you need to substitute the starting value of the series into the expression, then continue substituting the next value in the pattern until you reach the ending value. Finally, add up all of the results to get the sum of the series.

What is the purpose of using sigma notation?

Sigma notation is used to represent long or infinite series in a more concise and organized way. It allows for easier evaluation of these series and also helps to identify patterns and relationships between the terms.

What are some common patterns in sigma notation?

Some common patterns in sigma notation include arithmetic series, geometric series, and factorial series. Arithmetic series have a constant difference between terms, geometric series have a constant ratio between terms, and factorial series have a factorial term in the expression.

How can I practice evaluating expressions using sigma notation?

There are many online resources and practice problems available to help you practice evaluating expressions using sigma notation. You can also create your own series and practice evaluating them using sigma notation.

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