TITLE: Can you replace $x$ with $x^2$ for any Maclaurin series?
QUESTION [6 upvotes]: Let's say I have a Maclaurin series for some function $f(x)$, and to find Maclaurin series $f(x^2)$ can I just substitute $x^2$ for each term Maclaurin series for $f(x)$?

REPLY [5 votes]: If $$f(x)=\sum_{n\ge0}a_nx^n \text{ for }\vert x \vert < R,$$ then $$f(x^k)=\sum_{n\ge0}a_nx^{nk} \text{ for }\vert x^k \vert < R\text{, i.e.} \vert x \vert <R^{1/k}$$