Shifting the Expansion Point of a Power Series

If we have a function that has been expanded in a power series about the point \(x = x_{0}\)

\begin{equation}
f(x) = \sum\limits_{n=0}^{\infty} a_{n} (x – x_{0})^{n}
\end{equation}

and if the radius of convergence of this series is non-zero, then we can generate a power series expansion about a new point \(x = x_{1}\) if this point is in the radius of convergence of the original power series. The new series is [1]

\begin{equation}
f(x) = \sum\limits_{k=0}^{\infty} b_{k} (x – x_{1})^{k}
\end{equation}

\begin{equation}
b_{k} = \sum\limits_{n=0}^{\infty}
\begin{pmatrix}
n+k \\
k
\end{pmatrix}
a_{n+k} (x_{1} – x_{0})^{n}
\end{equation}

While this is clearly not practical for paper and pencil solutions, it does appear that it could be useful for computer applications. If you had an algorithm for determining the coefficients of the original power series and you wanted to generate many new power series with different expansion points, then this formulation would allow one to do this by only have to store the coefficients of the original power series. One can imagine having a series of expansion points, \(x_{1}s\), running a loop over them, and using this algorithm to generate different representations of the given function as power series.

[1] Theory and Application of Infinite Series by Konrad Knopp.