Complex Power Series

Many principles of Infinite Series and their Series Conversion Tests can be directly carried over to complex series.

Behavior as $n\to \infty$

• A complex series $\sum \_{n=0}^{\infty }=a\_0+a\_1z+...$ diverges if the sequence of partial sums $s\_n=a\_0+a\_1z+...$ has a limiting value $S=u\_s\_n(x,iy)+v\_s\_n(x,iy)$. Where $u\_n\to u$ and $v\_n\to v$.

Convergence Tests

Absolute Convergence Test for Complex Series

If a series $\sum \_{n=0}^{\infty }=a\_n{z}^n$ converges absolutely, that is its absolute counterpart $\sum \_{n=0}^{\infty }=|a\_n{z}^n|$ converges, then the series $\sum \_{n=0}^{\infty }=a\_n{z}^n$ converges itself.

Convergence Theorem

The conversion of a complex series can behave the following way

1. Converges for all $z$
2. Converges for $|z|<R$ (circle) and diverges for $|z|>R$
3. Converges only at $z=0$ Because of the polar interpretation of Complex Numbers this region is a circle centered at the origin with radius $R$, called the convergence circle.