An infinite sequence $\left\{z_1,z_2,z_3 \ldots\right\}$ of complex numbers has a limit $z$ if, for each positive number $\varepsilon$, there exists a positive integer $n_0$ such that \begin{eqnarray}\label{seq} \left|z_n-z\right|< \varepsilon \quad \text{whenever}\quad n > n_0. \end{eqnarray}
Geometrically, this means that for sufficiently large values of $n$, the points $z_n$ lie in any given $\varepsilon$ neighborhood of $z$ (Figure 1). Since we can choose $\varepsilon$ as small as we please, it follows that the points $z_n$ become arbitrarily close to $z$ as their subscripts increase. Note that the value of $n_0$ that is needed will, in general, depend on the value of $\varepsilon$.
The sequence $\left\{z_n\right\}_{n=1}^{\infty}$ can have at most one limit. That is, a limit $z$ is unique if it exists. When that limit exists, the sequence is said to converge to $z$; and we write \begin{eqnarray*} \lim_{n\rightarrow \infty} z_n=z \end{eqnarray*} If the sequence has no limit, it diverges.
Theorem 1: Suppose that $z_n=x_n+iy_n$ ($n=1,2,3,\ldots $) and $z=x+iy$. Then \begin{eqnarray}\label{teoseq01} \lim_{n\rightarrow \infty} z_n=z \end{eqnarray} if and only if \begin{eqnarray}\label{teoseq02} \lim_{n\rightarrow \infty} x_n=x\quad \text{and}\quad \lim_{n\rightarrow \infty} y_n=y. \end{eqnarray}
To prove this theorem, we first assume that conditions (\ref{teoseq02}) hold. That is, there exist, for each $\varepsilon>0$, positive integers $n_1$ and $n_2$ such that \[ |x_n-x|<\frac{\varepsilon}{2}\quad \text{whenever} \quad n>n_1 \] and \[ |y_n-y|<\frac{\varepsilon}{2}\quad \text{whenever} \quad n>n_2. \] Hence if $n_0$ is the larger of the two integers $n_1$ and $n_2$, \[ |x_n-x|<\frac{\varepsilon}{2}\quad \text{and} \quad |y_n-y|<\frac{\varepsilon}{2} \quad \text{whenever} \quad n > n_0. \] Since \[ |(x_n+iy_n)-(x+iy)|=|(x_n-x)+(y_n-y)|\leq |x_n-x|+|y_n-y|, \] then \[ |z_n-z|< \frac{\varepsilon}{2}+\frac{\varepsilon}{2} \quad \text{whenever} \quad n > n_0. \] Therefore condition (\ref{teoseq01}) holds.
Conversely, if we start with condition (\ref{teoseq01}), we know that for each positive number $\varepsilon$, there exists a positive integer $n_0$ such that \[ |(x_n+iy_n)-(x+iy)|<\varepsilon \quad \text{whenever} \quad n>n_0. \] However \[ |x_n-x|\leq |(x_n-x)+(y_n-y)|=|(x_n+iy_n)-(x+iy)| \] and \[ |y_n-y|\leq |(x_n-x)+(y_n-y)|=|(x_n+iy_n)-(x+iy)|. \] Consequently \[ |x_n-x|<\varepsilon \quad \text{and} \quad |y_n-y|<\varepsilon \quad \text{whenever} \quad n > n_0. \] Therefore, conditions (\ref{teoseq02}) are satisfied. $\blacksquare$
An infinite series \begin{eqnarray}\label{series01} \sum_{n=1}^{\infty}z_n=z_1+z_2+z_3+\cdots \end{eqnarray} of complex numbers converges to the sum $S$ if the sequence \begin{eqnarray}\label{partialsum} \sum_{n=1}^{N}z_n=z_1+z_2+z_3+\cdots +z_N\quad(N=1,2,3,\ldots) \end{eqnarray} of partial sums converges to $S$; we then write $$\sum_{n=1}^{\infty}z_n=S.$$ Note that since a sequence can have at most one limit, a series can have at most one sum. When a series does not converge, we say that it diverges.
Theorem 2: Suppose that $z_n=x_n+iy_n$ ($n=1,2,3,\ldots $) and $S=X+iY$. Then \begin{eqnarray}\label{teo01} \sum_{n=1}^{\infty} z_n=S \end{eqnarray} if and only if \begin{eqnarray}\label{teo02} \sum_{n=1}^{\infty} x_n=X\quad \text{and}\quad \sum_{n=1}^{\infty} y_n=Y. \end{eqnarray}
To prove this theorem, we first write the partial sums (\ref{partialsum}) as \begin{eqnarray}\label{teo03} S_N = X_N+iY_N, \end{eqnarray} where \[ X_N = \sum_{n=1}^{N}x_n \quad \text{and}\quad Y_N = \sum_{n=1}^{N}y_n. \] Now statement (\ref{teo01}) is true if and only if \begin{eqnarray}\label{teo04} \lim_{N\rightarrow \infty} S_N = S; \end{eqnarray} and, in view of relation (\ref{teo03}) and Theorem 1 on sequences, limit (\ref{teo04}) holds if and only if \begin{eqnarray}\label{teo05} \lim_{N\rightarrow \infty} X_N=X\quad \text{and}\quad \lim_{N\rightarrow \infty} Y_N=Y. \end{eqnarray} Limits (\ref{teo05}) therefore imply statement (\ref{teo01}), and conversely. Since $X_N=X$ and $Y_N=Y$ are partial sums of the series (\ref{teo02}), the theorem is proved. $\blacksquare$
This theorem can be useful in showing that a number of familiar properties of series in calculus carry over to series whose terms are complex numbers.
Property 1: If a series of complex numbers converges, the $n$-th term converges to zero as $n$ tends to infinity.
It follows from Property 1 that the terms of convergent series are bounded. That is, when series (\ref{series01}) converges, there exists a positive constant $M$ such that $$|z_n| \leq M \; \text{ for each positive integer } n.$$Another important property of series of complex numbers that follows from a corresponding property in calculus is the following.
Property 2: The absolute convergence of a series of complex numbers implies the convergence of that series.
Recall that series (\ref{series01}) is said to be absolutely convergent if the series \begin{eqnarray*}\label{series02} \sum_{n=1}^{\infty}|z_n|=\sum_{n=1}^{\infty}\sqrt{x^2_n+y^2_n}\quad \quad(z_n=x_n+iy_n) \end{eqnarray*} of real numbers $\sqrt{x^2_n+y^2_n}$ converges.
To establish the fact that the sum of a series is a given number $S$, it is often convenient to define the remainder $\rho_N$ after $N$ terms, using the partial sums: \begin{eqnarray*}\label{series03} \rho_N=S-S_N \end{eqnarray*} Thus $S=S_N+\rho_N$. Now, since $|S_N-S|=|\rho_N-0|$, then a series converges to a number $S$ if and only if the sequence of remainders tends to zero.
Example: With the aid of remainders, it is easy to verify that \begin{eqnarray*} \sum_{n=0}^{\infty}z^n=\frac{1}{1-z}\quad\text{whenever}\quad |z|< 1 \end{eqnarray*} We need only recall the identity $$1+z+z^2+\cdots+z^n=\frac{1-z^{n+1}}{1-z}$$ to write the partial sums \begin{eqnarray} S_N(z)=\sum_{n=0}^{\infty}z^n=1+z+z^2+\cdots+z^{N-1}\quad\quad(z\neq 1) \end{eqnarray} as $$S_N(z)=\frac{1-z^N}{1-z}.$$ If $$S(z)=\frac{1}{1-z}$$ then $$\rho_N(z)=S(z)-S_N(z)=\frac{z^N}{1-z}\quad\quad(z\neq 1).$$ Thus $$\left|\rho_N\right|=\frac{|z|^N}{|1-z|}\rightarrow 0\quad\text{only when}\quad |z|<1.$$ In this case, it is clear that the remainders $\rho_N$ tend to zero when $|z|<1$ but not when $|z|\geq 1$.
The series introduced in the previous example \begin{eqnarray*} \sum_{n=0}^{\infty}z^n=\frac{1}{1-z}\quad\text{whenever}\quad |z|< 1 \end{eqnarray*} is known as the geometric series.
Use the following applet to explore this series. Drag the point $z$ around. Observe what happens when it is inside, outside or on the border of the unit circle. Drag the slider to show the partial sum.
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