My book says that the Cauchy-Hadamar theorem reads: Consider a power serie [itex]\sum_{n=0}^{\infty} a_n x^n[/itex]. The radius of convergence is given by(adsbygoogle = window.adsbygoogle || []).push({});

[tex]r = \frac{1}{\lim_{n \rightarrow \infty} |a_n|^{1/n}}[/tex]

But on mathworld, it is enounced as follow (notice that it's the UPPER limit).

Which is considerably different, because for the serie

[tex]\sum_{n=0}^{\infty}\frac{x^{2n}}{5^n}[/tex]

for exemple, which can be rewritten as a power serie of the form

[tex]\sum_{n=0}^{\infty}a_n x^n[/tex]

where a_n = 0 if n is even and a_n = 5^(-n/2) if n is odd, we have that [itex]\lim_{n \rightarrow \infty} |a_n|^{1/n}[/itex] does not exist because the subsequence a_n where n is even converges towards 0 and the subsequence a_n where n is odd converges towards 5^(-1/2). So the limit doesn't exist.

On the other hand, the upper limit of the sequence |a_n|^{1/n} is obviously 5^(-1/2) since |a_n| is decreasing.

So the two theorems are not equivalent. :grumpy:

Who's right?

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# Cauchy-Hadamard theorem

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