Express a series in sigma notation

In summary, the conversation revolves around finding the sigma notation for the given series 3, 4, 6, 10, 18... between r=0 and r=infinity. The participants discuss the pattern of the series and suggest using 2r or 2+2r as the function to represent the series. There is also confusion about whether the question is asking for a sequence or a series, but it is ultimately clarified that the question is asking for a general closed form for the numbers in the series.
  • #1
whatisreality
290
1

Homework Statement


I've been given the series 3,4,6,10,18... and asked to express as the ∑ar, between r=0 and r = infinity.

Homework Equations

The Attempt at a Solution


Well, I can see a pattern! The difference between terms doubles every time. I'm having difficulty expressing this mathematically though... I think it should be 3 plus some function of r. Is there a technique for this? All we've been told is to do it 'by inspection'.
 
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  • #2
You're on the right track. It goes up by 1, then 2, 4, 8. What pattern does that seem like? Can you think of a function that increments in that manner
 
  • #3
2r might work?
So that would make the series 3+2r, except then it doesn't work for r=0...
Maybe 2+2r instead. That works! :)
 
Last edited:
  • #4
whatisreality said:

Homework Statement


I've been given the series 3,4,6,10,18... and asked to express as the ∑ar, between r=0 and r = infinity.

Homework Equations

The Attempt at a Solution


Well, I can see a pattern! The difference between terms doubles every time. I'm having difficulty expressing this mathematically though... I think it should be 3 plus some function of r. Is there a technique for this? All we've been told is to do it 'by inspection'.

The differences double, so the terms are 3, 3+1, 3+1+2, 3+1+2+4, 3+1+2+4+8,...
 
  • #5
Ray Vickson said:
The differences double, so the terms are 3, 3+1, 3+1+2, 3+1+2+4, 3+1+2+4+8,...
That's doing it the hard way! Reminds me of that story about John Nash solving the back-and-forth fly problem by summing the series...
 
  • #6
haruspex said:
That's doing it the hard way! Reminds me of that story about John Nash solving the back-and-forth fly problem by summing the series...

Might not be the hard way if the question was not really stated correctly by the OP; have we not seen that many times before in this forum? If the OP gave the correct statement, then, of course, you are right; but if the question really wanted an expression for ##a_r## in sigma notation, then what I outlined would go part way to the solution.
 
  • #7
Ray Vickson said:
Might not be the hard way if the question was not really stated correctly by the OP; have we not seen that many times before in this forum? If the OP gave the correct statement, then, of course, you are right; but if the question really wanted an expression for ##a_r## in sigma notation, then what I outlined would go part way to the solution.
I'm confused. I want an expression for ar using sigma notation. I thought that's what my question asked. What does my question actually mean then? As in, I thought I stated the problem correctly, AND the problem was I want an expression using sigma notation.
 
  • #8
whatisreality said:
I'm confused. I want an expression for ar using sigma notation. I thought that's what my question asked. What does my question actually mean then? As in, I thought I stated the problem correctly, AND the problem was I want an expression using sigma notation.

Well, if you want to express the general (##r##th) term ##a_r## of your sequence in sigma notation, the method I suggested in Post # 4 points the way. Just try to translate that into sigma notation.
 
  • #9
whatisreality said:
I've been given the series 3,4,6,10,18... and asked to express as the ∑ar, between r=0 and r = infinity.

Asked to express what as the ∑ar?
 
  • #10
Ray Vickson said:
Might not be the hard way if the question was not really stated correctly by the OP; have we not seen that many times before in this forum? If the OP gave the correct statement, then, of course, you are right; but if the question really wanted an expression for ##a_r## in sigma notation, then what I outlined would go part way to the solution.
Yes, it's not very clear. Because it referred to the given numbers as a series (which to me implies summation), not a sequence, I was looking for a general closed form for the numbers ##a_0 = 3, a_1=4, ##... so that the series sum can be written ##\Sigma a_r##.
But you are interpreting it as a sequence, which must then 'unsummed' into a series. ##a_0 = 3, a_0+a_1=4, a_0+a_1+a_2=6##... That makes it seem a strange question to me, quite apart for the terminology.
 
  • #11
I'm not entirely sure how to differentiate between those two cases. I think I meant the first option though, as in a0=3, a1=4 etc. Sorry for a lack of clarity!
 

1. What is sigma notation?

Sigma notation is a mathematical shorthand used to express a series in a more compact form. It is represented by the Greek letter sigma (Σ) and allows us to write a sum of terms in a concise way.

2. How do you write a series in sigma notation?

To write a series in sigma notation, we use the following format: Σi=kn ai, where i is the index variable, k is the starting value, n is the ending value, and ai represents the terms in the series.

3. What is the purpose of using sigma notation?

The main purpose of using sigma notation is to simplify and condense the notation of a series, making it easier to read and work with. It also allows us to express infinite series and series with a large number of terms in a more manageable way.

4. How do you evaluate a series given in sigma notation?

To evaluate a series given in sigma notation, we substitute the values of the index variable i into the expression ai and sum up all the resulting terms. This will give us the final value of the series.

5. Can a series be written in multiple ways using sigma notation?

Yes, a series can be written in multiple ways using sigma notation. For example, we can change the starting and ending values, the index variable, or the expression for the terms. However, all these notations will represent the same series and will have the same final value when evaluated.

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