# How do i evaluate the convergence

• supercali
In summary, to evaluate the convergence of a Fourier series, one can use the average error method to determine the number of terms needed for it to converge to less than X% difference. This involves solving with n variables and then solving with n+1 variables and comparing the difference between the two. Once the difference is less than X%, the number of terms can be determined. This method works because each individual component in the series has less value towards the total solution than the previous one.
supercali
got Fourier series as a result of solving a PDE. how do i evaluate the converg. using average error in order to determine the # of terms needed for it to converge to less than X%?

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Well brute force is Solve with n vars then solve with n+1 vars and compare |n-(n+1)|/|n+1| once that is less than X% difference you can do that. Since each individual component should have less value towards the total solution than the previous one.

ChaseRLewis said:
Well brute force is Solve with n vars then solve with n+1 vars and compare |n-(n+1)|/|n+1| once that is less than X% difference you can do that. Since each individual component should have less value towards the total solution than the previous one.

oops was thinking about this I mistyped it's |f(n)-f(n+1)|/|f(n+1)| where f is your pde function and n equivalent to the number of components. meant that before but was rushing.

## 1. What is convergence?

Convergence refers to the process of a sequence or function approaching a specific value or point. In scientific terms, it is often used to describe the behavior of a system as it approaches a stable state or equilibrium.

## 2. How do I know if a sequence or function is convergent?

In order to determine the convergence of a sequence or function, you can use various mathematical tests such as the ratio test, the root test, or the comparison test. These tests involve evaluating the behavior of the sequence or function as the input value increases, and determining if it approaches a fixed value.

## 3. What is the difference between absolute and conditional convergence?

Absolute convergence describes a sequence or function that converges regardless of the order of the terms, while conditional convergence only occurs when the terms are arranged in a specific order. In other words, absolute convergence guarantees convergence, while conditional convergence may not always occur.

## 4. Can a divergent sequence or function ever converge?

No, a divergent sequence or function will never converge. Divergence means that the sequence or function is not approaching a specific value or point, and therefore cannot be considered convergent.

## 5. What is the importance of evaluating convergence in scientific research?

Evaluating convergence is crucial in scientific research because it allows us to determine the stability and behavior of a system. Convergence can also help us make predictions about the future behavior of a sequence or function, and can provide insights into the underlying principles and mechanisms of a system.

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