# root test applied to power series

When I apply the root test to power series I know that

$$R= \frac{1}{\limsup_{n\rightarrow\infty}{\sqrt[n]{|c_n|}}}$$

So if I get $1$ as the result of the limit the radius of convergence of the power series is $R=1$?

Same question also for the ratio test.

I'm confused because when I study the behaviour of numerical series with these criteria and I get 1 as a result it means that they are not applicable.

The prove of the ratio test given on Wikipedia though seems to me clear in indicating that in the power series case $1$ is not a problematic result.

• Yes and yes.${}{}{}{}{}$ – DonAntonio Feb 9 '18 at 11:45
• Yeeeeees don't confused the Radius of convergence and the criteria for simple series – Guy Fsone Feb 9 '18 at 11:51
• Thank you I'll keep in mind – Gitana Feb 9 '18 at 16:07

If you apply the ratio test to $\sum c_n z^n$: $$(\limsup_{n\rightarrow\infty}\root n\of{|c_n|})|z| = \limsup_{n\rightarrow\infty}\root n\of{|c_n z^n|} < 1$$ implies that the series is convergent for $$|z| < \frac{1}{\limsup_{n\rightarrow\infty}{\sqrt[n]{|c_n|}}}.$$