Rate of Convergence for g(x) Limit at x=0

No, the conversation remains the same. Can you provide a summary of the conversation? In summary, the conversation discusses the rate of convergence for a function g(x) and the confusion surrounding it. The professor's explanation is confusing, and the person asking for advice is struggling to understand it. They also ask for the definition of rate of convergence and mention that it was given as an example problem by a PhD professor. Ultimately, the conversation ends without a clear understanding of the rate of convergence for g(x).
  • #1
ductiletoaste
5
0
Ok i have a question I am have been stuck on for a while.
g(x) = f(x)^(1/3) and f(x)^(1/3) = (e^x-cos(x)-x-x^2)^(1/3)

What is the rate of convergence for lim x->0 of g(x)?

Now to make it easier i took the taylor poly of the crazy function to degree 3. Which is (x^3)/6

The part I am confused about is what our prof told us in class...
lim h->0 of G(h) = 0, and lim h->0 of F(h) = L
We say that F(h) converges to L with a Rate of Convergence O(G(h)).

So what is the rate of convergence? and what is o(G(h))?
Thank you for any direction you can provide!
 
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  • #2
Can anyone give me any advice?
 
  • #3
The description you gave of the professor's explanation is very confusing.
 
  • #4
ur telling me. I have been lost and that's a not a description its a direct quote from one of his lectures! But for the sake of helping me figure this out let's forget the whole confusing explantion. how would u normally find the Rate of convergence of g(x) = f(x)^(1/3) and f(x)^(1/3) = (e^x-cos(x)-x-x^2)^(1/3)

What is the rate of convergence for lim x->0 of g(x)?
 
  • #5
Could you define "rate of convergence"?
 
  • #6
no offense but if u don't know what rate of convergence is then u can't hope to answer this question.
 
  • #7
I've been a mathematician all my life. I never heard that particular term used in an absolute sense, only relative to something else, like x3 -> 0 faster than x2 when x -> 0.
 
  • #8
Well this was an example problem given to me by a Phd professor. So i can assure you there is nothing wrong with the question.
 
  • #9
We say that F(h) converges to L with a Rate of Convergence O(G(h)).

So what is the rate of convergence? and what is o(G(h))?
The first statement says that |F(h)-L| -> 0 at the same rate as G(h)->0.
The second statement means that |F(h)-L| -> 0 faster than G(h) -> 0.

Did you edit the statement after first posting?
 
Last edited:

1. What is the rate of convergence for g(x) at the limit x=0?

The rate of convergence for g(x) at the limit x=0 refers to how quickly the function g(x) approaches its limit as x gets closer and closer to 0. It is a measure of how rapidly the values of g(x) converge or approach the limit at x=0.

2. How is the rate of convergence calculated?

The rate of convergence is typically calculated using mathematical formulas or algorithms specific to the function g(x). It may involve taking the limit of the ratio of the difference between consecutive terms of the sequence and the difference between the limit and the last term of the sequence.

3. What factors affect the rate of convergence for g(x) at x=0?

The rate of convergence for g(x) at x=0 can be affected by several factors, such as the initial value of g(x) at x=0, the behavior of g(x) near x=0, and the order of the function g(x) (i.e. whether it is linear, quadratic, etc.). Additionally, the choice of algorithm or method used to calculate the rate of convergence can also impact the results.

4. Why is the rate of convergence important?

The rate of convergence is important because it provides insight into the behavior of a function at a specific limit. It can also be used to compare the convergence of different functions or algorithms, and can help determine the efficiency and accuracy of numerical methods.

5. Can the rate of convergence be negative?

No, the rate of convergence cannot be negative. It is typically expressed as a positive number or as a limit, and represents the speed at which a function approaches its limit. A negative rate of convergence would imply that the function is diverging instead of converging.

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