Convergence of Power Series without Recursion Relation

In summary, the conversation discusses finding the interval of convergence for a given power series, which does not have a recursion relation. The person suggests using the ratio test, but the other person points out that knowledge about the coefficients is necessary to determine the appropriate test. The purpose of finding the convergence is to solve a nonlinear ODE using a power series solution. The potential radius of convergence is determined to be when |x| is less than 1, and any convergence test can be used. The conversation ends with a request for the ODE and its answer for more accurate results.
  • #1
Winzer
598
0

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?
 
Last edited:
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  • #2
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
So no?
 
  • #4
Winzer said:
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.
 
  • #5
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
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
ok but it is ugly:

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

Related to Convergence of Power Series without Recursion Relation

1. What is a power series?

A power series is an infinite series of the form $\sum_{n=0}^{\infty} a_nx^n$, where $a_n$ are coefficients and $x$ is the variable. It is a type of mathematical representation that can be used to describe various functions.

2. How do you determine if a power series converges?

The convergence of a power series depends on the value of $x$. One method for determining convergence is the ratio test, where the series converges if the limit of the absolute value of the ratio of consecutive terms is less than 1. Another method is the root test, where the series converges if the limit of the nth root of the absolute value of the nth term is less than 1.

3. What is the interval of convergence for a power series?

The interval of convergence is the range of values for $x$ where the power series converges. It can be determined by using the ratio or root test, and is typically expressed as an interval from $-R$ to $R$, where $R$ is known as the radius of convergence.

4. Are there any other tests for convergence of power series?

Yes, there are other tests such as the integral test, comparison test, and limit comparison test. These tests can be used in certain cases when the ratio or root test may not be applicable.

5. Can a power series diverge?

Yes, a power series can diverge if the limit of the ratio or root test is greater than 1 or undefined. It can also diverge if the series does not satisfy the conditions of any of the convergence tests. In these cases, the power series does not have a defined sum.

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