Is there an inverse of Summation?

In summary: I don't really think there is one universal method. Some things work in one occasion but not in the other. I think the best you can do is to consider a specific function and try to work it out for that.
  • #1
cmcraes
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Say for some general function f(x), and g(x) = ∑x=0 f(x) (assuming function is defined)
Is there a way to find the zeroes of g(x)? Is there any relationship between the zeroes of f(x) and g(x)? Sorry if this question is poorly asked, i just began learning about summations and infinite series.
Thanks
 
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  • #2
cmcraes said:
Say for some general function f(x), and g(x) = ∑x=0 f(x)

This makes no sense.

First a minor point, but your summation index is ##x##, which is an unusual choice. So I assume you sum over the naturals?

Second, your summation index is ##x## so it shouldn't be used outside the summation. Setting ##g(x)## equal to this makes little sense to me.

Indeed, by definition we can write

[tex]\sum_{x=0}^{+\infty} f(x) = f(0) + f(1) + f(2) + f(3) + ...[/tex]

So your equality

[tex]g(x) = \sum_{x=0}^{+\infty} f(x)[/tex]

means

[tex]g(x) = f(0) + f(1) + f(2) + f(3) + ...[/tex]

which is probably not what you want.
 
  • #3
What if we assume N is being used in the function f(x)
And we reset g(x) (I'm on mobile right now so I can't use symbols) to equal

G(x) = Summation from N=0 to +Infinity of f(x)?
 
  • #4
cmcraes said:
What if we assume N is being used in the function f(x)
And we reset g(x) (I'm on mobile right now so I can't use symbols) to equal

G(x) = Summation from N=0 to +Infinity of f(x)?

Now ##f(x)## is independent from ##N##. So you're just adding a bunch of constants. Is this your intention?
 
  • #5
How about a function f(nx)?
(Maybe I just need to go learn more about Infinite series and Functions)
 
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  • #6
cmcraes said:
How about a function f(nx)?
(Maybe I just need to go learn more about Infinite series and Functions)

So you're considering

[tex]g(x) = \sum_{n=0}^{+\infty} f(nx)[/tex]

?
 
  • #7
Say the function f(x) is (off the top of my head): x^3/(1-n)^x

How would we go about finding the zeroes g(x) of the sum of From n=0 to +infinity? Or am I asking all the wrong questions?
 
  • #8
cmcraes said:
Say the function f(x) is (off the top of my head): x^3/(1-n)^x

How would we go about finding the zeroes g(x) of the sum of From n=0 to +infinity? Or am I asking all the wrong questions?

I don't really think there is one universal method. Some things work in one occasion but not in the other. I think the best you can do is to consider a specific function and try to work it out for that.
 
  • #9
OKay, Thanks!
 

1. What is the inverse of summation?

The inverse of summation is known as the difference operator, also denoted by the symbol Δ. It is the opposite of summation and is used to calculate the difference between consecutive terms in a series.

2. How is the inverse of summation calculated?

The inverse of summation is calculated by subtracting the previous term from the current term in a series. For example, if we have the series 2, 4, 6, 8, the inverse of summation would be 2, 2, 2, as we are subtracting 2 from each term to get the next term in the series.

3. Does every series have an inverse of summation?

Not every series has an inverse of summation. This is because the series must have a constant difference between consecutive terms for the inverse of summation to be defined. If the series has a varying difference between terms, the inverse of summation cannot be calculated.

4. What is the relationship between summation and its inverse?

Summation and its inverse are inverse operations. This means that they undo each other and can be used to cancel each other out. For example, if we have the series 2, 4, 6, 8 and apply the inverse of summation, we will get the original series back.

5. Are there any practical applications of the inverse of summation?

Yes, the inverse of summation has many practical applications in mathematics and science. It is commonly used in calculus and statistics to calculate changes in values over time. It is also used in engineering and economics to analyze data and make predictions.

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