Convergence of Power Series without Recursion Relation

[Winzer]

Homework Statement

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

Homework Equations

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?

 

[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?

 

[Winzer]

So no?

 

[Dick]

So no?

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.

 

[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.

 

[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 ! :)

 

[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