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

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


  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?


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 $.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.