For which values of $ z$ does the series
$ \sum_{n\equal{}1}^{\infty}c_1c_2\cdots c_n z^n$
converge, provided that $ c_k>0$ and
$ \sum_{k\equal{}1}^{\infty} \frac{c_k}{k}<\infty$ ?
To determine the values of \( z \) for which the series \(\sum_{n=1}^{\infty} c_1 c_2 \cdots c_n z^n\) converges, we need to analyze the behavior of the terms \( c_1 c_2 \cdots c_n z^n \).

1. **Consider the general term of the series:**
   \[
   a_n = c_1 c_2 \cdots c_n z^n
   \]

2. **Apply the ratio test to determine the radius of convergence:**
   The ratio test states that a series \(\sum a_n\) converges if:
   \[
   \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| < 1
   \]
   For our series, we have:
   \[
   \left| \frac{a_{n+1}}{a_n} \right| = \left| \frac{c_1 c_2 \cdots c_{n+1} z^{n+1}}{c_1 c_2 \cdots c_n z^n} \right| = |c_{n+1} z|
   \]

3. **Find the limit of the ratio:**
   \[
   \lim_{n \to \infty} |c_{n+1} z|
   \]
   For the series to converge, we need:
   \[
   \lim_{n \to \infty} |c_{n+1} z| < 1
   \]
   This implies:
   \[
   |z| < \frac{1}{\limsup_{n \to \infty} c_{n+1}}
   \]

4. **Use the given condition \(\sum_{k=1}^{\infty} \frac{c_k}{k} < \infty\):**
   Since \(\sum_{k=1}^{\infty} \frac{c_k}{k}\) converges, by the comparison test, \(c_k\) must tend to zero as \(k\) tends to infinity. Therefore, \(\limsup_{n \to \infty} c_n = 0\).

5. **Determine the radius of convergence \(R\):**
   Since \(\limsup_{n \to \infty} c_n = 0\), we have:
   \[
   \frac{1}{\limsup_{n \to \infty} c_n} = \infty
   \]
   Thus, the radius of convergence \(R\) is infinite, meaning the series converges for all \(z\).

The final answer is \(\boxed{\text{all } z \in \mathbb{C}}\).