TITLE: Can the the radius of convergence increase due to composition of two power series?
QUESTION [10 upvotes]: When composing power series, is the radius of convergence the minimum of that of the individual series, or is it like for multiplication and addition of power series where the resultant radius of convergence may be larger than for the individual series?
If the radius of convergence is the minimum as described, then is the interval of convergence the intersection of the two individual intervals of convergence, or is behaviour at the end-points not guaranteed (just like it isn't when differentiating a power series to obtain another)?

REPLY [3 votes]: How about an example:
$$
\frac x{1-x}\circ\frac x{1+x} = x\quad?
$$