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In one of my tasks, I needed to calculate the radius of convergence for the power series $\sum_{k=0}^{\infty}\frac{z^{3k}}{2^k}$.

Using the ratio test for general series, I get $\sqrt[3]{2}$ as the radius which is correct according to the book. Now I wanted to try to get the radius with the Cauchy-Hadamard formula for power series.

With the series in this form I think that I can't use the Cauchy-Hadamard formula so I've written it a bit different, like this:

$$ \sum_{k=0}^{\infty}a_k\,z^k,\quad a_k= \begin{cases} \frac{1}{2^k},& k=3j,\;j\in\mathbb{N}_0\\ 0,&\text{otherwise} \end{cases} $$

When using this, I get $$\limsup_{k\to \infty}\sqrt[k]{|a_k|} = \frac{1}{2}$$

So according to Cauchy-Hadamard the radius of convergence would be $2$ and this is wrong. Where is my mistake?

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1 Answer 1

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When $k=3j,$ $a_{k}=\frac{1}{2^j},$ not $\frac1{2^k}.$

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