Complex sequence

A complex sequence is a list $z_1,z_2,z_3,\dots$ of complex numbers. It converges to a limit $L\in \mathbb{C}$ if $|z_n-L|\to 0$ as $n\to \infty$.

Reduction to real sequences

Because

$|z_n-L|=\sqrt{(\mathrm{Re}{z}_n-\mathrm{Re}L{)}^2+(\mathrm{Im}{z}_n-\mathrm{Im}L{)}^2},$

the complex sequence converges to $L$ if and only if both the real parts $\mathrm{Re}{z}_n$ and the imaginary parts $\mathrm{Im}{z}_n$ converge to $\mathrm{Re}L$ and $\mathrm{Im}L$ respectively. So everything from real-variable convergence theory carries over componentwise.

Examples

• $z_n=1/n+i/n^2\to 0$. Both components go to $0$.
• $z_n=(n+i)/n=1+i/n\to 1$.
• $z_n=(1+i/n{)}^n\to e^i$. Proved rigorously via the Taylor series in power series theory.
• $z_n=i^n$ cycles $i,-1,-i,1,i,\dots$ without approaching anything — divergent.

The last example is unusual in real-variable terms: the sequence is bounded ($|i^n|=1$ for all $n$) but doesn’t converge. Boundedness is necessary but not sufficient for convergence.

Cauchy criterion

$z_n$ converges iff for every $\epsilon >0$ there is $N$ such that $|z_m-z_n|<\epsilon$ for all $m,n\ge N$. Cauchy and convergent are equivalent in $\mathbb{C}$ as in $\mathbb{R}$ — $\mathbb{C}$ is complete with the usual metric.

Connection to series and the complex derivative

Convergence of complex sequences is the foundation for power series, Taylor series, and Laurent series in complex analysis. The Complex derivative is itself a limit — the limit of a difference quotient as $\Delta z\to 0$ in the plane, which is a sequential-limit statement requiring the same value along every direction of approach. That “every direction” condition is what makes complex differentiability much more restrictive than real differentiability.

A diagnostic limit that fails

The function $f(z)=z/|z|$ has no limit at $0$. Approach along the positive real axis ($z=x,x\to 0^{+}$): $f=x/x=1$. Approach along the positive imaginary axis ($z=iy$): $f=iy/y=i$. Different directional limits, so ${\lim }_{z\to 0}f(z)$ does not exist. This kind of “approach-dependent” failure is the engine of why $\bar{z}$, $\mathrm{Re}(z)$, $|z|$ fail the complex derivative test.