Proof

(related to Proposition: Complex Exponential Function)

From the ratio test for absolutely convergent complex series it follows that

\[\left|\frac{\frac{z^{n+1}}{(n+1) ! }}{\frac{z^n}{n ! }}\right|=\frac{|z|}{n+1}\le\frac 12.\]

Therefore, the complex exponential series

\[\sum_{n=0}^\infty\frac{z^n}{n!}\]

is an absolutely convergent complex series for every complex number \(z\in\mathbb C\), where \(n!\) denotes the factorial of the index \(n\).


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References

Bibliography

  1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983