Is there an inverse of Summation?

[cmcraes]

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

 

[micromass]

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

$$\sum_{x=0}^{+\infty} f(x) = f(0) + f(1) + f(2) + f(3) + ...$$

So your equality

$$g(x) = \sum_{x=0}^{+\infty} f(x)$$

means

$$g(x) = f(0) + f(1) + f(2) + f(3) + ...$$

which is probably not what you want.

 

[cmcraes]

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)?

 

[micromass]

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?

 

[cmcraes]

How about a function f(nx)?
(Maybe I just need to go learn more about Infinite series and Functions)

 

[micromass]

How about a function f(nx)?
(Maybe I just need to go learn more about Infinite series and Functions)

So you're considering

$$g(x) = \sum_{n=0}^{+\infty} f(nx)$$

?

 

[cmcraes]

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?

 

[micromass]

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.

 

[cmcraes]

OKay, Thanks!