Interval of convergence?

  • #1
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 in to the series.

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


my "Interval of Convergence" is (-∞,∞). SO now lets 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.
 

Answers and Replies

  • #2
115
2
Sum of series exists only for numbers from that interval. If you take other numbers you don't get any "answer".
 
  • #3
296
0
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.
 
  • #4
HallsofIvy
Science Advisor
Homework Helper
41,833
963
"radius of convergence" and "interval of convergence" are two different things. If we have a (real valued) power series of the form [itex]\sum a_n(x- a)^n[/itex] 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 [itex]\sum a_n (z- a)^n[/itex], where a, z, and every [itex]a_n[/itex] 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.
 
  • #5
Bacle2
Science Advisor
1,089
10
Some nice properties happen within the radius of convergence ,like the fact that you

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

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