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In definition 5.1 ,
we defined a sequence in
to be a function
. Since we are identifying
with a subset of
, every
sequence in
is also a sequence in
, and all of our results for complex
sequences are applicable to real sequences.

** 7.1**
**Notation ()**
I will say `` consider the sequence

" or `` consider the
sequence

" to mean `` consider the sequence

such that

for all

". The arrow

is
read `` maps to".

** 7.2**
**Definition (Geometric sequence.)**
For each

, the sequence

is called the

*geometric sequence with ratio* .

I will often represent a sequence in
by a polygonal line with vertices
.
The two figures below represent geometric sequences with
ratios
and
respectively.

** 7.3**
**Definition (Geometric series.)**
If

, then the sequence

is called the

*geometric series with ratio*
.

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 and the figure representing
.

** 7.4**
**Entertainment.**
Describe the apparent similarity between the figure for

and the
figure for

. Then prove that this similarity is
really present for all

.

** 7.5**
**Definition (Constant sequence.)**
For each

, let

,
denote the constant sequence

; i.e.,

.

** Next:** 7.2 Convergence
** Up:** 7. Complex Sequences
** Previous:** 7. Complex Sequences
** Index**