# Solving to the exact

1. Aug 10, 2010

### darkside00

Is is possible to solve every real integral and come up with an actual solution? perhaps we may have not just found the methods of doing so. Or is it a must to use approximations(series/sequences) to do so?
Or is there a way to reverse numerical numbers to come up with a function?

2. Aug 10, 2010

### Hurkyl

Staff Emeritus
It really depends on what you mean by "actual solution".

Taylor series aren't approximations, by the way. You can truncate the series after a finite number of terms to produce an approximation, but the Taylor series itself is an exact representation of an analytic function -- at least within its radius of convergence.

3. Aug 10, 2010

### darkside00

right. I guess all series can be expressed as functions with n to infinite within its convergence.

So, perhaps the real question is could we establish functions as a non series, or do we need them?

Last edited: Aug 11, 2010
4. Aug 11, 2010

### HallsofIvy

I think you need to formulate a precise question. What do you mean by "establish functions as a non series"?

5. Aug 11, 2010

### FallenRGH

I think what he means is to define a function analytically without using a series expansion, and I believe the answer you are looking for is that, short of defining a function as the integral of another function(for example, the logarithmic integral function), we currently cannot define some functions short of a series expansion.