What is the Physical Interpretation of Infinite Series?

[matqkks]

Is there a physical intrepretation of infinite series?
Is there a picture that will explain what is meant by infinite series or is there a tangible application of this concept.

 

[JesseC]

An infinite series is a mathematic construct that pops up in physics all the time.

To actually analyse them, they must either converge to a single value, or be taken to an appropriate number of terms.

Infinite series are commonly used to approximate functions, but only the first 2 or 10 or however many terms of the series will be used. It would be impossible to evaluate an infinite number of terms in a finite amount of time!

 

[l'Hôpital]

Infinite series are commonly used to approximate functions, but only the first 2 or 10 or however many terms of the series will be used. It would be impossible to evaluate an infinite number of terms in a finite amount of time!

I'd be careful with such statements. Zeno's paradox might beg to differ.

 

[samuelarnold]

Infinite sequences and series continue indefinitely. You can make terms in infinite series as much as long.

 

[rojas70]

If you go to wikipedia and lookup the definition for a http://en.wikipedia.org/wiki/Taylor_series" , you'll see a couple of figures that give a good visual understanding of how suming the derivatives of a function, tend to that function.

I, on the other hand have the question of what exactly is the implication of using variable "a" in the Taylor Series, where "a" is the real or complex value representing the neighborhood for the series.

What happens when a is smaller or bigger?

 

[JesseC]

Woah... the OP was almost a year ago :D

 

[JesseC]

Ah just realized you also asked a question, the 'a' in that Wikipedia article denotes the x co-ordinate about which you are approximating the function f(x)... So to make it bigger or smaller simply means to shift the region you are approximating.

Suppose you wanted to approximate cos(x) about the point x=5, then a=5.

 

[scottbekerham]

Why isn't there a chapter or sections in mathematical physics textbooks that talks about the methods of summing series . all books shows only how to discover if a series is convergent or not

 

[netzweltler]

Is there a physical intrepretation of infinite series?
Is there a picture that will explain what is meant by infinite series or is there a tangible application of this concept.

Isn't Zeno's paradoxes the best example for a physical application of infinite series?

 

[raxAdaam]

*An* interesting physical interpretation I've seen (i.e. of a particular infinite series) was the case of stacking rectangular blocks on top of each other.

Although difficult to explain in words, the question that motivates the resultant infinite series is "is it possible to stack identical blocks on top of each other such that, at some point, the n-th block does not overlap with the first (e.g. if the blocks have length 1, and the coordinate system is drawn such that the first block extends from the origin to x=1, is it possible to stack additional blocks on top of that first block such that eventually a block will be located entirely to the right of x=1, and is, so to speak, "not directly supported").

One can model this situation w/ some basic knowledge of centre of mass and the result is an infinite series. The question then becomes - does the series converge or diverge? If the series converges, then there is a finite horizontal extent to which stacked blocks can stretch out. If the series does not converge, then (in principle), one could start with a single block and create a stack where the top block is arbitrarily far away (in horizontal distance) from the bottom block.

In this case, it turns out the series diverges. However, the height of the stack grows much faster than its horizontal extent - both go to infinity, but the latter much more slowly than the former.

Not sure if this is the kind of thing you were thinking of, but I personally found it a very enlightening approach to limits (& I suspect there are many others ...).

Cheers,

Rax

 

[SteveL27]

Woah... the OP was almost a year ago :D

Zeno must be right then :-)

 

[micromass]

This post is already 2 years old.