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Revised Absolute Convergence

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I am not so satisfied with this the following definition.1

\[ e^{i\theta} := \cos \theta + i \sin \theta \]

I remembered the proof for convergence of

\[ e^{x} := \sum_{i = 0}^{\infty} \frac{x^i}{i!} \]

for real numbers. I didn’t know if this can be extended to complex numbers. Therefore, I thought about the absolute convergence of complex-valued series. It’s expected that many proofs are similar to their real counterparts, such as the result that absolute convergence implies convergence. In real numbers, this result makes use of the Triangle Inequalty and Cauchy Convergence Criterion, and the key step is

\[ \abslr{\sum_{k = m + 1}^{n} a_k} \le \sum_{k = m + 1}^{n} \abs{a_k}. \]

Since the proof of the above statement for real numbers requires Bolzano–Weierstrass Theorem, which is about the sequential compactness of sequences of real numbers, I was stuck at this point.

Finally, I read another book, which said that if $(z_n)$ is a Cauchy sequence, and $\forall n \in \N$, $u_n := \Re(z_n)$ and $v_n := \Im(z_n)$, $\forall \varepsilon > 0, \exists N \in \N$ such that $\forall m,n \le N$,

\begin{align*} \abs{u_n - u_m} &\le \abs{z_n - z_m} < \varepsilon, \\ \abs{v_n - v_m} &\le \abs{z_n - z_m} < \varepsilon. \end{align*}

Then $(u_n)$ and $(v_n)$ are real-valued Cauchy sequences, which are convergent.2 This guarantees the convergence of $(z_n)$ in the complex plane.

To establish the absolute convergence of $\exp z$, we need the root test. The proofs can be borrowed from their counterparts in the set of real numbers. Ahlfors leaves the proof for $\sqrt[n]{n!} \to \infty$ to readers. I find Dan’s proof pretty easy.

  1. Brown, J. W., Churchill, R. V., & Lapidus, M. (1996). Complex variables and applications (Vol. 7). (pp. 17). New York: McGraw-Hill. ↩

  2. Ahlfors, L. (1979). Complex analysis. (pp. 34). Auckland: McGraw-Hill. ↩

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