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Loser66
 one year ago
Find the radius of convergence:
\(\sum_{n=0}^{\infty} z^{n!}\)
Please, help
Loser66
 one year ago
Find the radius of convergence: \(\sum_{n=0}^{\infty} z^{n!}\) Please, help

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dan815
 one year ago
Best ResponseYou've already chosen the best response.0hmmm greater than 1 and less than 1

Loser66
 one year ago
Best ResponseYou've already chosen the best response.0\(\sum_{n=0}^\infty z^{n!}= \sum_{n=0}^\infty a_m z^{n!}\) where \(a_m = \begin{cases}0 ~~if ~~ m \neq n!\\m ~~if ~~m = n!\end{cases}\)

Loser66
 one year ago
Best ResponseYou've already chosen the best response.0but I don't know how to deal with the first z, since \(sum_{n=0}^\infty z^{n!}= z^{0!} + z^{1!}+z^{2!}+...... = \color{red}{z} + \sum something we can handle\)

Loser66
 one year ago
Best ResponseYou've already chosen the best response.0the red z kills me. hehehe ...

Loser66
 one year ago
Best ResponseYou've already chosen the best response.0if n = 1 to infinitive, that is we don't have that red z, then apply Dirichlet's test, we can have the radius of convergence is \(R= \dfrac{1}{lim sup \sqrt[m] {a_m}}=1\)
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