7.1 Some Examples.

In definition 5.1 , we defined a sequence in $\mbox{{\bf C}}$ to be a function $f\colon\mbox{{\bf N}}\to\mbox{{\bf C}}$. Since we are identifying $\mbox{{\bf R}}$ with a subset of $\mbox{{\bf C}}$, every sequence in $\mbox{{\bf R}}$ is also a sequence in $\mbox{{\bf C}}$, and all of our results for complex sequences are applicable to real sequences.

7.1   Notation ($\mapsto$) I will say `` consider the sequence $n\mapsto 2^n$" or `` consider the sequence $f\colon n\mapsto 2^n$" to mean `` consider the sequence $f\colon\mbox{{\bf N}}\to\mbox{{\bf C}}$ such that $f(n)=2^n$ for all $n\in\mbox{{\bf N}}$". The arrow $\mapsto$ is read `` maps to".

7.2   Definition (Geometric sequence.) For each $\alpha\in\mbox{{\bf C}}$, the sequence

$$n\mapsto\alpha^n$$

is called the geometric sequence with ratio $\alpha$.

I will often represent a sequence $f$ in $\mbox{{\bf C}}$ by a polygonal line with vertices $f(0), f(1), f(2), \cdots$. The two figures below represent geometric sequences with ratios $${ {{1+i}\over 2}}$$ and $${{{2+i}\over 3}}$$ respectively.

7.3   Definition (Geometric series.) If $\alpha\in\mbox{{\bf C}}$, then the sequence $${ g_\alpha\colon n\mapsto\sum_{j=0}^n\alpha^j}$$ is called the geometric series with ratio $\alpha$.

$$g_\alpha=\{1,1+\alpha,1+\alpha+\alpha^2,1+\alpha+\alpha^2+\alpha^3,\cdots\}$$

Figure b shows the geometric series corresponding to the geometric sequences in figure a. If you examine the figures you should notice a remarkable similarity between the figure representing $\{\alpha^n\}$ and the figure representing $${\{\sum_{j=0}^n\alpha^j\}}$$.

7.4   Entertainment. Describe the apparent similarity between the figure for $\{\alpha^n\}$ and the figure for $${\{\sum_{j=0}^n\alpha^j\}}$$. Then prove that this similarity is really present for all $\alpha\in\mbox{{\bf C}}\backslash\{1\}$.

7.5   Definition (Constant sequence.) For each $\alpha\in\mbox{{\bf C}}$, let $\tilde\alpha$, denote the constant sequence $\tilde\alpha\colon n\mapsto\alpha$; i.e., $\tilde\alpha=\{\alpha,\alpha,\alpha,\alpha,\cdots\}$.