Find the value of the positive constant \(c_0\) such that the series given below

\[\displaystyle \sum_{n=0}^{\infty} \dfrac{n!}{{(cn)}^n}\]

diverges for \(0 < c < c_0\) and converges for \(c > c_0\).

**Clarification**: \(e \approx 2.71828\) denotes the Euler's number.

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