https://file.aiquickdraw.com/mathbot/file/1725697282779hf2jpaax.jpg

The problem shown in the image asks to find the radius of convergence of the series:

\[
\sum_{n=1}^{\infty} \frac{(z^n)}{\cos(n)}
\]
where \( z = x + iy \) is a complex number.

### Steps to Solve:

To determine the radius of convergence of a complex series of the form \(\sum a_n z^n\), we use the formula for the radius of convergence \(R\):

\[
\frac{1}{R} = \limsup_{n \to \infty} \sqrt[n]{|a_n|}
\]

For the given series, the general term is:

\[
a_n = \frac{1}{\cos(n)}
\]

#### Key Points:

1. \(\cos(n)\) oscillates between -1 and 1, meaning that \(\frac{1}{\cos(n)}\) is bounded for \(n\), but \(\cos(n)\) is never zero (except for specific values where \(n\) is not a real number).
2. Since \(a_n\) does not tend to 0 as \(n\) approaches infinity, the series does **not converge** for any nonzero \(z\).

Therefore, the radius of convergence for this series is:

\[
R = 1
\]

Hence, the correct answer is:

\[
\boxed{1}
\]

### Would you like more details or have any questions?

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Here are five related questions to deepen understanding:

1. How is the radius of convergence related to the \(\limsup\) of coefficients?
2. What is the importance of \( \cos(n) \)'s behavior in this series?
3. How would the series behave if \( \cos(n) \) approached zero?
4. Can the radius of convergence ever be infinite?
5. What role does the imaginary part \( iy \) of \( z \) play in convergence?

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**Tip:** When analyzing complex series, always check the behavior of the series' coefficients and apply the root test or ratio test to determine convergence.

Find the radius of convergence of the series ∑ (z^n)/cos(n), where z = x + iy.

This page explains how to calculate the radius of convergence for the series ∑ (z^n)/cos(n), where z = x + iy. Using the root test, the problem demonstrates the steps to find the radius of convergence. Suitable for undergraduate students studying applied mathematics.

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