Power series vs. taylor series

In summary, a Taylor series is a specific type of power series formed from a function and can either equal or approximate the function. It is possible for the Taylor series of an infinitely differentiable function to not equal the function itself, as shown in the example of f(x).
  • #1
ehilge
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Hey all,
So I have a physics final coming up and I have been reviewing series. I realized that I'm not quite sure on what the differences are between a Taylor series and a power series. From what I think is true, a taylor series is essentially a specific type of power series. Would it be correct to say that a power series is just the sum of a random sequence whereas a Taylor series is also a sum of a sequence but one that can approximate a function. Anything I'm missing here? Is there anything else special about a taylor series?
Thanks for your help!
 
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  • #2
The Taylor's series of a function is a power series formed in a particular way from that function. Of course, if a power series is equal to a function (not "approximate a function") then that power series is the Taylor's series for the function.

On the other hand, it is possible that the Taylor's series for an infinitely differentiable function is NOT equal to the function at all. An example is
[tex]f(x)= \left{\begin{array}{c} e^{-1/x^2} if x\ne 0 \\ 0 if x= 0\end{array}\right[/tex]
f is infinitely differentiable and the value of every derivative at x= 0 is 0 so its Taylor's series about 0 is just
[tex]\sum_n=0^\infty \frac{0}{n!}x^n[/tex]
which, of course, converges to 0 for all x but is equal to f only at x= 0.
 

1. What is the difference between a power series and a Taylor series?

A power series is a representation of a function as an infinite sum of terms with increasing powers of a variable. A Taylor series is a specific type of power series that is centered around a specific point and uses the derivatives of the function at that point to determine the coefficients of the terms.

2. When should I use a power series versus a Taylor series?

Power series can be used to approximate a function over a larger range of values, while Taylor series are best used to approximate a function near a specific point. If you need to approximate a function over a small interval, a Taylor series may be more accurate. For larger intervals, a power series may be a better choice.

3. Can a power series and a Taylor series be used interchangeably?

No, power series and Taylor series are not interchangeable. Taylor series are a specific type of power series that require a specific point of expansion and use the derivatives of the function at that point to determine the coefficients. Power series can be used to approximate a function over a larger range of values, while Taylor series are best used to approximate a function near a specific point.

4. How are the coefficients in a power series and a Taylor series determined?

The coefficients in a power series are determined by the general formula for the series, while the coefficients in a Taylor series are determined by the derivatives of the function at a specific point. The more derivatives used in a Taylor series, the more accurate the approximation will be.

5. Are there any limitations to using power series or Taylor series to approximate a function?

Yes, both power series and Taylor series are only useful for approximating smooth, continuous functions. If a function has sharp corners or discontinuities, these methods will not provide accurate results. Additionally, the accuracy of the approximation depends on the number of terms used in the series, so it may not be exact for all values of the function.

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