How to do expansion as power series of any random function?

In summary, the conversation discusses the topic of expanding a function as a power series and the difficulty of finding clear and thorough explanations on the internet. The solution to this problem is to use Taylor Expansion, which is a mathematical method for representing a function as an infinite sum of terms.
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darussiaman
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How to do expansion as power series of any random function??

The template doesn't really apply because this is a general question rather than a specific problem.

If I am given some function, how do I expand it as a power series? For the past two semesters of my physics degree, I have been seeing this done repeatedly. Yet I still don't know how to do it, really. It isn't taught anywhere! The textbooks we use assume that we know how it works, and also that apparently it's quite simple. But none of our required classes specifically go over this topic; supposedly it was taught way back in Calc 2 but that was years ago, in high school, and as far as I remember it wasn't really taught from this angle anyway... the focus was more about how to work with and what to do with series when you have them rather than how to create a series when you don't have one.

So I want to be like those authors in the textbooks: I'm given some function, bam, I spit out the equivalent power series. How do I do that? Any source on the internet where this is explained clearly and thoroughly? (Obviously wikipedia is not helpful at all...)

Actually, the specific problem I was trying to deal with is this: expand sqrt(1 + epsilon) as a power series.

Thanks a lot
 
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1. How do I determine the coefficients in a power series expansion?

The coefficients in a power series expansion can be determined by repeatedly differentiating the function and evaluating at the expansion point. Alternatively, if the function can be expressed as a known power series, then the coefficients can be determined by comparing terms.

2. What is the radius of convergence for a power series expansion?

The radius of convergence for a power series expansion is the distance from the expansion point where the series converges. It can be determined by using the ratio test on the series.

3. Can any function be expanded as a power series?

No, not all functions can be expanded as a power series. The function must be analytic, meaning it must have derivatives of all orders at every point within its domain.

4. How do I know if a power series expansion is accurate?

A power series expansion is accurate if the difference between the value of the function and the value of the power series at a given point approaches 0 as the number of terms in the series approaches infinity. This can be checked by evaluating the remainder term of the series.

5. Is there a general formula for expanding a function as a power series?

There is no general formula for expanding any function as a power series. However, there are specific methods for expanding certain types of functions, such as polynomial, trigonometric, and exponential functions.

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