Sum of Series w/ 2's & 3's: Find the Total

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In summary: Thanks!In summary, the sum of the series 1 + 1/2 + 1/3 + 1/4 + 1/6 + 1/8 + 1/9 + 1/12 + ..., when written in sigma notation, is 1 + 1/2 + 1/4 + 1/8 + 1/16 + ... + 1/3 + 1/9 + 1/27 + ... + 1/6 + 1/12 + 1/18 + 1/24 + ... - 2.
  • #1
iNCREDiBLE
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find the sum of the series 1 + 1/2 + 1/3 + 1/4 + 1/6 + 1/8 + 1/9 + 1/12 + ...,
where the terms are the reciprocals of the positive integers whose only prime factors are 2's and 3's.

Here's my work so far:

1 + 1/2 + 1/3 + 1/4 + 1/6 + 1/8 + 1/9 + 1/12 + ... =
1 + 1/2 + 1/4 + 1/8 + 1/16 + ... + 1/3 + 1/9 + 1/27 + ... + 1/6 + 1/12 + 1/18 + 1/24 + ... =
1 + 1/2 + 1/4 + 1/8 + 1/16 + ... + 1 + 1/3 + 1/9 + 1/27 + ... + 1 + 1/6 + 1/12 + 1/18 + 1/24 + ... - 2

Those first two are geometric series. I don't know what to do with the third one.
 
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  • #2
Try writing it in sigma notation. (Σ)
 
  • #3
Hurkyl said:
Try writing it in sigma notation. (Σ)

:confused:
My problem is to find the value of 1 + 1/6 + 1/12 + 1/18 + 1/24 + 1/32 + 1/48 + 1/54...
 
  • #4
That sum is no easier to calculate than the original sum. (In fact, I would be able to figure out the original sum faster than the one in your last post)

There are other ways to see how to approach this problem, but IMHO, writing it in sigma notation makes the right step much clearer.

In general, it's good practice to write sums in sigma notation anyways.
 
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  • #5
Hurkyl said:
That sum is no easier to calculate than the original sum. (In fact, I would be able to figure out the original sum faster than the one in your last post)

There are other ways to see how to approach this problem, but IMHO, writing it in sigma notation makes the right step much clearer.

In general, it's good practice to write sums in sigma notation anyways.

Well then my problem is "sigma notation". :cry:
 
  • #6
Do you know how to write a simpler sum in sigma notation? Like 1 + 1/2 + 1/4 + 1/8 + 1/16 + ...?
 
  • #7
Hurkyl said:
Do you know how to write a simpler sum in sigma notation? Like 1 + 1/2 + 1/4 + 1/8 + 1/16 + ...?

Yes.
[tex]\sum_{n=0}^\infty \frac{1}{2^n}[/tex]
 
  • #8
Ok, good. Next question: what's the general form of a number whose prime factors are only 2 and 3?
 
  • #9
Hurkyl said:
Ok, good. Next question: what's the general form of a number whose prime factors are only 2 and 3?

That's [tex]2^n3^m[/tex]
 
  • #10
So does that suggest how to write the sum of the reciprocals of every positive integer whose prime factors can only be 2's and 3's in sigma notation?
 
  • #11
Hurkyl said:
So does that suggest how to write the sum of the reciprocals of every positive integer whose prime factors can only be 2's and 3's in sigma notation?

Indeed. Is it going to be a double sum?
 
  • #12
Yes, it is!


Technical note: actually, the double sum is technically different from the original sum. However, the sum turns out to be absolutely convergent (the easiest way to see this fact is that it has no negative terms!), and this sort of "rearrangement" is perfectly legal with absolutely convergent sums, meaning that they'll give the same answer.
 
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  • #13
Hurkyl said:
Yes, it is!

And is the correct answer 3? :blushing:
 
  • #14
Sounds right. Well done! By the way, don't miss the technical note I edited into my last post.

By the way, now that you know how the problem is solved, it may be instructive to try and figure out how to do it without sigma notation. Then again, it might not be instructive too, I don't know. :smile:
 
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  • #15
Hurkyl said:
Sounds right. Well done! By the way, don't miss the technical note I edited into my last post.

By the way, now that you know how the problem is solved, it may be instructive to try and figure out how to do it without sigma notation. Then again, it might not be instructive too, I don't know. :smile:

Thanks for the technical note. Had forgot it. o:)
Without sigma notation? I have no clue at all :tongue:
 
  • #16
You had it as a double sum right? With a bit of rearranging you should be able to write it as a product of 2 infinite sums, which I suppose you could write without sigma notation if you liked. I'd suggest you try this as the product form you'll get is quite cute and worth the effort.
 
  • #17
shmoe said:
You had it as a double sum right? With a bit of rearranging you should be able to write it as a product of 2 infinite sums, which I suppose you could write without sigma notation if you liked. I'd suggest you try this as the product form you'll get is quite cute and worth the effort.

Yes, that's exactly what I did.
 

1. What is the formula for finding the sum of a series with 2's and 3's?

The formula for finding the sum of a series with 2's and 3's is:
S = (n/2) * (3 + 2n), where n is the number of terms in the series.

2. How do you find the total of a series with 2's and 3's?

To find the total of a series with 2's and 3's, you can use the formula:
S = (n/2) * (3 + 2n), where S represents the sum and n represents the number of terms in the series. Simply plug in the value of n and solve for S to find the total.

3. What is an example of calculating the sum of a series with 2's and 3's?

For example, if we have a series with 5 terms, the sum would be:
S = (5/2) * (3 + 2*5) = (5/2) * (3 + 10) = (5/2) * 13 = 32.5
Therefore, the total of the series with 2's and 3's is 32.5.

4. Can this formula be used for series with other numbers besides 2's and 3's?

Yes, this formula can be used for series with any common difference between terms. In the formula, the number 2 represents the common difference between terms, so it can be replaced with any other number. However, the number 3 cannot be changed as it is a constant in the formula.

5. How can this formula be applied in real life or scientific contexts?

This formula can be used in various scientific fields, such as physics and biology, to calculate the total of a series with a constant or regular increase. For example, in physics, it can be used to find the total distance traveled by an object with a constant acceleration. In biology, it can be used to calculate the total population growth over a period of time with a constant rate of increase.

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