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.
$\endgroup$ 31 Answer
$\begingroup$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 $$|z| < \frac{1}{\limsup_{n\rightarrow\infty}{\sqrt[n]{|c_n|}}}.$$
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