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