Sequences and Series in the Complex Plane

Reading

Section 1.4, part I

Problems

Practice problems:

1. Show that \(z_n\to z\) if and only if \(\bar z_n\to \bar z\).
2. Suppose that \(|z| < 1\). Show that \(a_n = z^n\to 0\).
3. True or False: The series \(\sum_{n=0}^\infty a_n\) converges if and only if the series \(\sum_{n=0}^\infty \textrm{Re}(a_n)\) and \(\sum_{n=0}^\infty \textrm{Im}(a_n)\) both converge.
4. True or False: If \(\sum_{n=0}^\infty a_n\) converges then \(\sum_{n=0}^\infty \bar a_n\) also converges.
5. If \(\sum_{n=0}^\infty a_n\) converges, show that \(\sum_{n=0}^\infty a_n^2\) also converges.

Topics to know

1. Sequence \(z_n\) converges to \(z\) iff \(|z_n - z|\to 0\).
   • Alternative definition
   • Laws for convergent sequences
2. Sequence converges if and only if its real parts converge and its imaginary parts converge.
   • Key inequality: \(|\textrm{Re(z)}|,|\textrm{Im(z)}|\leq |z|\leq |\textrm{Re(z)}| + |\textrm{Im(z)}|\)
3. Cauchy sequences, for both real numbers and complex numbers.
   • Laws for Cauchy sequences
4. Convergent sequences are Cauchy.
5. Cauchy sequences of real numbers converge. Key steps:
   • Cauchy sequences are bounded.
   • Any sequence contains a monotone subsequence.
     • Key statement: There is an \(N\) such that for all \(M\geq N\) there is a \(K\geq M\) such that \(a_K\geq a_M\).
     • If that is true, then can build an increasing subsequence.
     • If it is false, then we can build a decreasing subsequence.
   • If a Cauchy sequence has a convergent subsequence then it converges.
6. Cauchy sequences of complex numbers converge.
   • Go through their real/imaginary parts.
7. Series of complex numbers. Review of results from Calculus 3. How do they carry over to complex numbers?
   • Definition of Convergent Series.
   • Divergence test.
   • Geometric series.
   • Definition of absolute/conditional convergence.
   • Alternating series test.
   • Absolute convergence implies conditional convergence.
   • Root and ratio tests.