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 via the Taylor series in power series theory.
• $z_n=i^n$ cycles $i,-1,-i,1,i,\dots$ without approaching anything, so it diverges.

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 underlies power series, Taylor series, and Laurent series. 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. The same approach-dependent failure is why $\bar{z}$, $\mathrm{Re}(z)$, $|z|$ flunk the complex-derivative test.