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How can I determine the convergence radius of the following power series:

$F(n,z):=\sum_{j=0}^nz^{3j^2}+5z^{j^3}$ ?

I've tried with the formulas

$ r=\frac{1}{\limsup_{n \to \infty} \sqrt{|a_n|}} $ and $r = \lim_{n \to \infty} |\frac{a_n}{a_{n+1}}|$ but hadn't any success yet.

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Assuming that you mean the series $\sum_{j=0}^\infty z^{3j^2}+5z^{j^3}$, then:

  • since it diverges when $z=1$, the radius of convergence is at most $1$;
  • since each series $\sum_{j=0}^\infty z^{3j^2}$ and $\sum_{j=0}^\infty5z^{j^3}$ converges when $\lvert z\rvert<1$ (by the ratio test), the sum converges when $\lvert z\rvert<$ and therefore the radius of convergence is at least $1$.

So, the radius of convergence is equal to $1$.

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