# Power Series expansion of an eigenvalue

• ExplosivePete
In summary, the conversation discusses the task of expanding the eigenvalue as a power series in epsilon, up to second order. The poster is familiar with power series but is unsure how to apply it in this context. Another user suggests looking for the term sqrt(1+4*epsilon^2) in the equation and developing it into a power series in epsilon, up to second order. The guidelines state that responding with "Dunno" is not acceptable.
ExplosivePete
1. ... Expand the Eigenvalue as a power series in epsilon, up to second order:
λ=[3+√(1+4 ε^2)]V0 / 2

## Homework Equations

I am familiar with power series, but I don't know how to expand this as one.[/B]

## The Attempt at a Solution

:[/B] I have played around with the idea of using known power series of functions such as e^x, yet I haven't found a way to make that useful.

Last edited:
Dunno is not good enough according to PF guidelines
But do I detect a term ##\sqrt{1+4\varepsilon^2}## in there ?

BvU said:
Dunno is not good enough according to PF guidelines
But do I detect a term ##\sqrt{1+4\varepsilon^2}## in there ?

Please review my post again. Hopefully it is both more legible and up to PF guidelines.

I still detect a term ##
\sqrt{1+4\varepsilon^2\;}## ! How would you develop that into a power series in ##\varepsilon##, up to second order ?

And the guidelines tell you 'Dunno' ('Don't know') isn't good enough...

## 1. What is a power series expansion of an eigenvalue?

A power series expansion of an eigenvalue is a mathematical representation of an eigenvalue as an infinite sum of terms, each of which is a polynomial of increasing degree. It is used to approximate the value of an eigenvalue in cases where it cannot be explicitly calculated.

## 2. How is a power series expansion of an eigenvalue derived?

A power series expansion of an eigenvalue is derived using Taylor's theorem, which states that any infinitely differentiable function can be represented as a power series. The coefficients of the power series are determined by taking derivatives of the function at a specific point.

## 3. What is the importance of a power series expansion of an eigenvalue?

A power series expansion of an eigenvalue is important because it allows for a more precise approximation of the eigenvalue, especially in cases where it cannot be explicitly calculated. It also provides insight into the behavior of the eigenvalue as the underlying variable changes.

## 4. What are the limitations of a power series expansion of an eigenvalue?

A power series expansion of an eigenvalue may not always converge, meaning that the infinite sum does not produce a finite value. In addition, it may not provide an accurate approximation for certain types of functions, such as those with singularities or discontinuities.

## 5. How is a power series expansion of an eigenvalue used in practical applications?

A power series expansion of an eigenvalue is used in various fields of science and engineering, such as physics, chemistry, and electrical engineering. It is particularly useful in solving differential equations, which are used to model many real-world phenomena, such as heat transfer and fluid flow.

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