Let $(z_n)_{n\in\mathbb{N}}$ be a complex sequence and $\alpha\in]0,\dfrac{\pi}{2}[$. For all $n\in\mathbb{N}$, the argument of $z_n$ is in $[-\alpha,\alpha]$. If $\sum z_n$ converges, prove that it's absolutely convergent.
Proving absolute convergence for complex sequences
-2
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real-analysis
sequences-and-series
convergence
1 Answers
1
Hint
Let $z_n=x_n+y_n\,i$. Since $|\arg{z_n}|\le\alpha<\pi/2$, $x_n\ge0$ and $|y_n|\le(\tan\alpha)\,x_n$.