$P(x)$ converges iff $Q(x)$ converges. Does this imply that both have the same radius of convergence? where $P(x)$ and $Q(x)$ are power series

Question

Let $P(x)$ and $Q(x)$ be two power series and $P(x)$ converges iff $Q(x)$ converges. Does this imply that both have the same radius of convergence?

score 1 · Answer 1 · answered Sep 30 '20 at 19:59

1

Yes.

Indeed, you can define the radius of convergence of a power series $\sum a_n z^n$ as the supremum of the $r \in \mathbb{R}_+$ such that $\sum a_n r^n$ converges.

answered Sep 30 '20 at 19:59

TheSilverDoe

29,720

Will you please elaborate? – Lucas Sep 30 '20 at 20:02

score 0 · Accepted Answer · answered Sep 30 '20 at 20:08

0

Call $R$ the radius of convergence for $P$, and $R^\prime$ the one for $Q$. For $|x|<R$, $P(x)$ converges, so by assumption $Q(x)$ also converges and this implies $|x|\leqslant R^\prime$. But $|x|$ can be chosen as close of $R$ as we want, so $R \leqslant R^\prime$. Doing the same thing for $Q$ instead of $P$, you get $R^\prime \leqslant R$. Finally, $R=R^\prime$

answered Sep 30 '20 at 20:08

what guarantees that $R\leq R'$ by choosing $|x|$ close to $R$ ? – Lucas Sep 30 '20 at 20:13
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Take $x_n$ such that $|x_n| <R$ and $|x_n| \to R$. Then for all $n$, $|x_n| \leqslant R^\prime$, and taking the limit gives $R \leqslant R^\prime$ – Sep 30 '20 at 20:19

$P(x)$ converges iff $Q(x)$ converges. Does this imply that both have the same radius of convergence? where $P(x)$ and $Q(x)$ are power series

2 Answers2