# Power Series Convergence

1. Aug 4, 2010

### Winzer

1. The problem statement, all variables and given/known data

Suppose I have the power series:
f(x) = A0 + A1 x +A2 x^2 ......An x^n
Where A0..An are numbers, there is no recursion relation.
Find the interval of convergence

2. Relevant equations

3. The attempt at a solution
Can I use ratio test?
How would I do this since there is no recursion relation for the A's?
Can I do this numerically?

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

### Dick

Of course you could try to use the ratio test. But you have to know SOMETHING about the A's. What do you if you don't know a recursion relation?

3. Aug 4, 2010

So no?

4. Aug 4, 2010

### Dick

So no, what? All I said is that you have to know something about the A's to figure out what test might work. If you don't know anything there is no way to answer. That's it.

5. Aug 4, 2010

### Winzer

Ok the reason I want to know convergence is because of the following:
I have a nonlinear ode. I want to find a power series solution for it. I throw in the series I stated and I get relations for the coefficents. I do have initial conditions. I need to know the radius of convergence.

6. Aug 5, 2010

### gomunkul51

What you have there is a Power Series, so generally speaking |x| must be less then 1 (you need to check separately for x=1).
This will give you the radius of convergence.

You CAN use the Ratio Test, you can use any other convergence test as well (they will give you the answer I wrote above).

P.S. show us you ODE and you answer, you will get more precise answers ! :)

7. Aug 5, 2010

### Winzer

ok but it is ugly:

$$y' = \left( c_1 x^3 - c_2 x^5 \right)$$
$$x' = \left( c_2 y^5 - c_1 y^3 \right)$$
c1 & c2 are free parameters