What does the Interval of Convergence for a power series tell me?

[BarringtonT]

So I know how to find the "Interval of Convergence" for a power series representation of a Function f(x).

But I Still don't know what that "Interval of Convergence" does for me other than I can choose a number in it and plug it into the series.

For Example e^{x}=\sum^{∞}_{n=0} \frac{x^n}{n!} ;when a=0;


my "Interval of Convergence" is (-∞,∞). SO now let's say i take the # 1 from my "Interval of Convergence" and place it in the series representation of e^x.

Then i would get back some answer , but what does that answer mean? besides the fact that I got an answer.

 

[szynkasz]

Sum of series exists only for numbers from that interval. If you take other numbers you don't get any "answer".

 

[Millennial]

Usually, it is called "radius of convergence" instead of "interval of convergence". A function that is dependent on a series only converges when its argument (in this case, x) has a smaller absolute value than the radius of convergence. The exponential function, the one you gave, is convergent for every x, and hence its radius of convergence is infinity.

 

[HallsofIvy]

"radius of convergence" and "interval of convergence" are two different things. If we have a (real valued) power series of the form \sum a_n(x- a)^n and I know that it has "radius of convergence", r, then I know that the series converges within the interval (a- r, a+ r), it "interval of convergence".

If we are dealing with complex valued power series, then the "radius of convergence" really is a radius If the series \sum a_n (z- a)^n, where a, z, and every a_n are complex numbers, has "radius of convergence" r, then it converges for all z in the interior of the disk with center at a and radius r.

 

[Bacle2]

Some nice properties happen within the radius of convergence ,like the fact that you

can do term-by-term integration and differentiation within it.