2.2 Complex Series

Let \((z_n)\) be a sequence of complex numbers. The form sum \(\, a_1 + a_2+ \cdots \,\) is called the series \(\,\displaystyle { \sum a_n}\).

\(\implies \,\) The series \(\displaystyle {\sum a_n}\) is said to converge to the sum(s) if the sequence of partial sums \((S_n)\) given by \[ S_n = a_1 + a_2 + \cdots + a_n = \sum ^n_{j=1}a_j\] Converges to the limit \((S)\) \(\quad S = \displaystyle {\sum ^{\infty }_{j=1}a_j}\)

\(\implies \,\) If the sequence \((S_n)\) does not converge, then we say that the series \(\displaystyle {\sum a_n}\) does not converge.

Proposition 2.8. \(\displaystyle {\sum z_n}\) converges if and only if \(\displaystyle {\sum \operatorname {Re}(z_n)}\) and \(\displaystyle {\sum \operatorname {Im} (z_n)}\) both converges.

Proof. Let \(z_n=x_n+iy_n\) and let \(S_N=\sum _{n=1}^{N}z_n\) be the partial sums. Since addition of complex numbers is componentwise, \[S_N=\sum _{n=1}^{N}x_n+i\sum _{n=1}^{N}y_n=A_N+iB_N ,\] where \(A_N\) and \(B_N\) are the partial sums of the real and imaginary series.

Convergence of a series is convergence of its sequence of partial sums, and a complex sequence converges precisely when its real and imaginary parts do — by the inequalities used in the previous proposition. So \(S_N\) converges if and only if \(A_N\) and \(B_N\) both converge, which is the statement.

This is what allows every convergence test for real series to be applied to a complex one, one component at a time. □

Properties

1.

(a): If \(\displaystyle {\sum z_n}\) converges then \(\, \lim \limits _{n\rightarrow \infty } z_n = 0\)
(b): there is a \(\, M\in \mathbb {R},\,\exists \, M\geq 0 \in \mathbb {R} \ni \left |z_n\right |\leq , \, \forall \, n \in \mathbb {N}\)

2.

If \(\displaystyle {\sum a_n}\) and \(\displaystyle {\sum b_n}\) are both convergent complex series then \(\displaystyle {\sum \big (a_n + C b_n\big )}\) is also convergent complex series \(\forall \,C \in \mathbb {R}\).

3.

Absolute Convergent
A series \(\displaystyle {\sum z_n}\) is said to be absolutely convergent if the series \(\displaystyle {\sum \left |z_n\right |}\) converges.
\(\implies \) Suppose that the real series \(\displaystyle {\sum \left |z_n\right |}\) converges then \(\displaystyle {\sum z_n}\) converges.

4.

Test for Convergence
We can apply comparison, ratio test and Cauchy’s \(n^{\text {th}}\) ratio test to test the convergence of \(\displaystyle {\sum z_n}\) and \(\displaystyle {\sum \left |z_n\right |}\) is convergent then \(\displaystyle {\sum z_n}\) is convergent by property (3)

[next] [prev] [prev-tail] [front] [up]

Questions on this section

Stuck on something here? Ask below and it stays attached to this topic.