Summation Sigma: How Can It Be Used?

In summary, the conversation discusses the use of the summation sign sigma in mathematics and its applications in various fields such as biology. The correct use of the notation is demonstrated through examples and the question of whether the amount of increment in each addition can be changed is raised. The conversation concludes with a clarification on how to find the sum on a calculator.
  • #1
disregardthat
Science Advisor
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(This is not a homework question!)

I have no education in this kind of math yet, but I wonder how many ways you are allowed to use the summation sign sigma. I can't seem to get a good explanation on google or wikipedia.

Since I like to try myself with tex, I will write an example of it here:

[tex]\sum_{k=3}^4 k^2[/tex] This will be a normal summation sign.

To see if I have got it right: [tex]k = 3[/tex] means that the k on the side of the sigma will start at 3, right?

If the sigma is raised to 4 like the one I have shown, it means that the k (3) will be added to a number 3+1, and then to 4+1, and then to 5+1. That the k is raised to the power of two, means that for each part of the serie, the number will be raised, like (3)^2 + (3+1)^2 + (4+1)^2... right?

I think this will be the same as: [tex]\sum_{k=3}^4 k^2 = 3^2 + 4^2 + 5^2 + 6^2 = 86[/tex]

Is this the correct use of the Summation?

-----------------------------

I have heard that it is used in biology to find out the number of cells that is being reproduced.
Let's say that we have a cell, and it has unlimited food, so it will reproduce in the rate of doubling each ten minutes. And we will watch it for one minute. That means that the cell and it's daughter cells will reproduce 60\10 = 6 times.
I found out that the only way that can be done is if you put it up like this:

[tex]\sum_{k=0}^6 2^k[/tex]

So I guess it would give us the right answer of how many cells that would be there.

[tex]\sum_{k=0}^6 2^k = 2^0 + 2^1 + 2^3 + 2^4 + 2^5 + 2^6 = 127[/tex]

The amount of cells that we will end up with, assuming none of them died, and assuming every cell reproduced itself each ten minutes, after one minute.

Is this a valid way of using the summation? If not, it should be :smile:
 
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  • #2
Jarle said:
(This is not a homework question!)

I have no education in this kind of math yet, but I wonder how many ways you are allowed to use the summation sign sigma. I can't seem to get a good explanation on google or wikipedia.

Since I like to try myself with tex, I will write an example of it here:

[tex]\sum_{k=3}^4 k^2[/tex] This will be a normal summation sign, right?

To see if I have got it right: [tex]k = 3[/tex] means that the k on the side of the sigma will start at 3 right?

If the sigma is raised to 4 like the one I have shown, it means that the k (3) will be added to a number 3+1, and then to 4+1, and then to 5+1. That the k is raised to the power of two, means that for each part of the serie, the number will be raised, like (3)^2 + (3+1)^2 + (4+1)^2... right?

I think this will be the same as: [tex]\sum_{k=3}^4 k^2 = 3^2 + 4^2 + 5^2 + 6^2 = 86[/tex]

Is this the correct use of the Summation?

No; In words, [tex]\sum_{k=3}^4 k^2 [/tex] means the "sum from k=3 to k=4 of k squared." So, this would be equal to 32+42.

I have heard that it is used in biology to find out the number of cells that is being reproduced.
Let's say that we have a cell, and it has unlimited food, so it will reproduce in the rate of doubling each ten minutes. And we will watch it for one minute. That means that the cell and it's daughter cells will reproduce 60\10 = 6 times.
I found out that the only way that can be done is if you put it up like this:

[tex]\sum_{k=0}^6 2^k[/tex]

So I guess it would give us the right answer of how many cells that would be there.

[tex]\sum_{k=0}^6 2^k = 2^0 + 2^1 + 2^3 + 2^4 + 2^5 + 2^6 = 127[/tex]

The amount of cells that we will end up with, assuming none of them died, and assuming every cell reproduced itself each ten minutes, after one minute.

Is this a valid way of using the summation? If not, it should be :smile:

This, however, is a correct use of the summation.

Note that the sigma notation is simply a shorthand way to write a long sum. To take a simple example, the sum 1+2+3+4+5+6 can be written in sigma notation as [tex]\sum_{n=1}^6n[/tex]
 
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  • #3
Thanks! I see that the power you raise the sum to is the point where it stops, and not the number of times you add the number in. I understand it.

Is it possible to make the amount of increasment in each addition to be lower or higherthan 1? Example: Is it correct to use this kind if summation:
[tex]\sum_{k=2}^2 \frac {4}{k} [/tex] And that would be: 2/4 + 3/4 + 4/4 = 2.25

EDIT: I meant this

[tex]\sum_{k=2}^4 \frac {k}{4} [/tex] And that would be: 2/4 + 3/4 + 4/4 = 2.25


Or is another way to make the amount of increasment lower, higher or in different numbers, for example [tex]pi[/tex]

Is this allowed for example, single yes and no answer is fine:

[tex]\sum_{k= \pi}^3k[/tex]
 
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  • #4
I'm not sure if I understand your question: That that use of the notation is fine, it would expand as [tex]\sum_{k=2}^8 \frac {4}{k} =\frac{4}{2} +\frac{4}{3}+\frac{4}{4}+\frac{4}{5}+\frac{4}{6}+\frac{4}{7}+\frac{4}{8}[/tex]
 
  • #5
Sorry, I mispelled the sum up there. But I got my answer. Thanks.

But not on the last one. I wonder if it is possible to use [tex]\pi[/tex] instead of 1 as the rising on each addition. I guess it would stand like this:

[tex]\sum_{k= \pi}^3k[/tex]


which would equal: [tex] = \pi + 2\pi + 3\pi = 6\pi[/tex]
 
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  • #6
Why not just write that one as:
[tex]\sum_{k=1}^{3}\pi{k}[/tex]
 
  • #7
arildno said:
Why not just write that one as:
[tex]\sum_{k=1}^{3}\pi{k}[/tex]

Thanks :smile:

And how excactly do I find this on my calculator, sorry for asking this much, but we are not learning this at school...

I find " Sum( " on my TI-84 Plus calculator, and I read that it should be put up like this: [tex]\sum[/tex](K,K,1,10) (for example) But I don't know how.
 
  • #8
How should I know how your calculator works??
Read the manual.
 
  • #9
This sum [tex]\sum_{k=2}^4 \frac {4}{k} [/tex] does not equal 2/4 + 3/4 + 4/4, it equal 4/2 + 4/3 + 4/4.

I think you may be a little confused here. The limits on the sum (here k=2,...,4) means the first term in the sum has k=2, and then the next terms increase the value of k by 1 each time.

The summation notation means that we sum over integers k, starting with k=the lower limit, and ending with k= upper limit. It does not make sense to set the bottom limit to pi (since this is not an integer). If you wish to write the sum [itex]\pi + 2\pi + 3\pi = 6\pi[/itex] using sigma notation, then use arildno's suggestion above.
 
  • #10
arildno said:
How should I know how your calculator works??
Read the manual.

I thought everyone got that calc, universal calc or something... Nevermind then, I'll sort it out somehow :smile:

Anyway, I understand it now, thank you.
 

1. What is summation sigma?

Summation sigma, or the Greek letter Σ, is a mathematical symbol used to represent the sum of a series of numbers or terms. It is commonly used in algebra and calculus to express the addition of multiple terms.

2. How can summation sigma be used in science?

Summation sigma can be used in science to represent the total sum of data points or measurements. For example, it can be used to calculate the total mass of a system by adding the masses of each individual component. It can also be used to calculate the average of a series of values by dividing the sum by the number of terms.

3. What are the key properties of summation sigma?

The key properties of summation sigma include commutativity, associativity, and distributivity. This means that the order in which the terms are added does not affect the result, terms can be grouped and added in any order, and the symbol can be distributed over multiplication or division.

4. Can summation sigma be used for infinite series?

Yes, summation sigma can be used for both finite and infinite series. In the case of infinite series, the symbol represents the limit of the sum as the number of terms approaches infinity. This is commonly used in calculus to evaluate integrals.

5. How can I simplify a summation sigma expression?

To simplify a summation sigma expression, you can use algebraic rules such as factoring, expanding, and combining like terms. You can also use known formulas, such as the sum of a geometric or arithmetic series, to simplify the expression. Additionally, you can use technology such as a graphing calculator or computer software to evaluate the expression numerically.

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