diff options
Diffstat (limited to 'pages/complex-analysis/one-complex-variable/power-series.md')
| -rw-r--r-- | pages/complex-analysis/one-complex-variable/power-series.md | 5 |
1 files changed, 3 insertions, 2 deletions
diff --git a/pages/complex-analysis/one-complex-variable/power-series.md b/pages/complex-analysis/one-complex-variable/power-series.md index 31793ab..4876d30 100644 --- a/pages/complex-analysis/one-complex-variable/power-series.md +++ b/pages/complex-analysis/one-complex-variable/power-series.md @@ -21,7 +21,8 @@ $a$ is the *center* of the series. {% enddefinition %} {% lemma %} -Suppose the Banach space valued power series $\sum_{n=0}^{\infty} x_n (z - a)^n$ converges for $z = a + w$. +Suppose the Banach space valued power series $\sum_{n=0}^{\infty} x_n (z - a)^n$ +converges for $z = a + w$. Then it converges absolutely for all $z$ with $\abs{z-a} < \abs{w}$. {% endlemma %} @@ -35,7 +36,7 @@ Then either - the series converges only for $z=a$ (formally $R=0$), or - there exists a number $0<R<\infty$ such that - the series converges absolutely whenever $\abs{z-a} < R$ + the series converges absolutely whenever $\abs{z-a} < R$ and diverges whenever $\abs{z-a} > R$, or - the series converges absolutely for all $z \in \CC$ (formally $R=\infty$). |