Express a series in sigma notation

  • #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'.
 

Answers and Replies

  • #2
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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! :)
 
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  • #4
Ray Vickson
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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
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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
Ray Vickson
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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
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
Ray Vickson
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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
Svein
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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
haruspex
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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!
 

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