moo5003 said:If f(x) has a power series: a_n(x-a)^n (centered at a)
what does the power series for f(2x) look like?
D H said:I would think he is after something like this:
[tex]f(2x) = \sum_{n=0}^{\infty}a_n(2x-a)^n = \sum_{n=0}^{\infty}a_n2^n\left(x-\frac a 2\right)^n =\sum_{n=0}^{\infty}b_n\left(x-\frac a 2\right)^n[/tex]
where [itex]b_n=a_n2^n[/itex]
A power series expansion is a mathematical concept used to represent a function as an infinite sum of terms, each with a different power of the independent variable. It is often used to approximate functions that are difficult to solve for directly.
A power series expansion allows us to approximate a function at any point by using a finite number of terms from the series. This can be useful in many areas of science, such as physics, engineering, and economics, where complex functions are often encountered.
The only difference is that in a power series expansion for f(2x), the independent variable is replaced by 2x in each term. This allows us to find a power series expansion for f(2x) by using the coefficients and exponents from the original power series for f(x).
Using a power series expansion for f(2x) can simplify calculations and make it easier to approximate the function at different points. It can also help us understand the behavior of the function, such as its rate of change and convergence.
Yes, there are some limitations. Power series expansions are only accurate for a certain range of values. Outside of this range, the approximation may not be accurate. Additionally, some functions may not have a power series expansion or may have a very complex one, making it difficult to use in practice.