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Goal: Find the radius of convergence for the following complex power series:

$$ \sum n^p z^n $$

Attempt:

  1. We have by Hadamard's formula for the radius of convergence that the complex power series $\sum a_n z^n$ converges if $|z| < R$ s.t.

    $$ R = \frac{1}{\underset{n \rightarrow \infty}{\limsup} |a_n|^{1/n}} $$

  2. Then we here have that $a_n = n^p$ for all $n \in \mathbb{N}$.

  3. Then applying Hadamard's formula we obtain that

    $$ R = \frac{1}{\underset{n \rightarrow \infty}{\limsup} |n^p|^{1/n}} $$

  4. Then since

    $$ \underset{n \rightarrow \infty}{\limsup} |n^p|^{1/n} = 1 $$

    no matter the value of $p$, we have that $R = 1$.

Is this correct?

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This is correct.

Here you can also use the more elementary approach using the d'Alembert lemma: there is convergence of the series as soon as

$$ 1 > \lim\frac {|(n+1)^pz^{n+1}|} {|n^pz^p|} = \lim \left(1 + \frac 1n\right)^p |z| = |z| $$ and divergence as soon as $$ 1 < \lim\frac {|(n+1)^pz^{n+1}|} {|n^pz^p|} = |z| $$ so $R = 1 $.

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