Infinite Series without an x term

[avocadogirl]

Infinite Series without an "x" term

What are the uses of an infinite series which does not include an "x" variable? If you are looking at an infinite series which sums terms based entirely upon each term's position in the series, or, a series which includes only the variable "n," how would that series be useful once evaluated in closed form?

Let's say you could evaluate a series, which only included "n," to the closed form of a trigonometric function. Would that be realistic since, the function would potentially have to be applied only to an "x" value of 1?

Or, where in physics would one need a purely numerical summation?

Thank you, sincerely.

 

[csprof2000]

Say that you're stacking charged disks.

Imagine, for the sake of simplicity, the force is such that the top disk is 1 m above the disk below it, that disk is 1/2 m above the disk below it, that disk is 1/4 m above the disk below it, etc.

Then I ask you how high the stack is.

 

[avocadogirl]

Thank you. I greatly appreciate the example. As simple and obvious that may have seemed, I failed to consider such a simple numerical application.

Adding to that, could anyone tell me about the numerical summation applied to a trigonometric function (closed form)?

 

[avocadogirl]

Does a series involving only "n" automatically lead one to consider Dirichlet?

 

[HallsofIvy]

This is really a very peculiar question. If there is no variable 'x' in a series, then it is a numerical series rather than a series of functions. And one normally studies numerical series before series of functions, defining the sum of a series of functions to be the function that gives the value of the numerical series you get for each specific value of x.