# Convergence radius of a perturbation series

• wdlang
In summary, the convergence radius of a perturbation series is a measure of how far away from the origin the series will converge, typically denoted by the symbol R. It is determined by analyzing the behavior of the coefficients of the series as the order of perturbation increases and can be found using mathematical techniques such as the ratio test or the root test. If the convergence radius is infinite, the series represents the exact solution to the problem. The convergence radius can change depending on the problem and the order of perturbation, and if the value of the variable is outside the convergence radius, the series will diverge and may require alternative methods to find a solution.
wdlang
i see people discussing the convergence radius of a perturbation series in the literature

i am really baffled

generally, one can only get the first few coefficients of a perturbation series

that is, the perturbation series is not known at all

how can one determine the convergence radius of the series at all?

what is the point of determine the radius if only a few first coefficients are available?

In general perturbation expansions aren't convergent, there will be a term which after that that series starts growing.

## 1. What is the convergence radius of a perturbation series?

The convergence radius of a perturbation series is a measure of how far away from the origin the series will converge. It is typically denoted by the symbol R and is the distance from the origin to the nearest point where the series diverges.

## 2. How is the convergence radius of a perturbation series determined?

The convergence radius of a perturbation series is determined by analyzing the behavior of the coefficients of the series as the order of perturbation increases. The radius can also be found by using mathematical techniques such as the ratio test or the root test.

## 3. What does it mean if the convergence radius of a perturbation series is infinite?

If the convergence radius of a perturbation series is infinite, it means that the series will converge for all values of the variable. This indicates that the series represents the exact solution to the problem and does not require any further approximations.

## 4. Can the convergence radius of a perturbation series change?

Yes, the convergence radius of a perturbation series can change depending on the problem being solved and the order of perturbation being used. It is possible for the radius to increase or decrease as the order of perturbation increases.

## 5. What happens if the value of the variable is outside the convergence radius?

If the value of the variable is outside the convergence radius, the perturbation series will diverge and will not provide an accurate solution to the problem. In this case, alternative methods such as numerical methods may be used to find a solution.

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