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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
![$n\mapsto 2^n$](img6.gif)
" or `` consider the
sequence
![$f\colon n\mapsto 2^n$](img7.gif)
" to mean `` consider the sequence
![$f\colon\mbox{{\bf N}}\to\mbox{{\bf C}}$](img3.gif)
such that
![$f(n)=2^n$](img8.gif)
for all
![$n\in\mbox{{\bf N}}$](img9.gif)
". The arrow
![$\mapsto$](img5.gif)
is
read `` maps to".
7.2
Definition (Geometric sequence.)
For each
![$\alpha\in\mbox{{\bf C}}$](img10.gif)
, the sequence
is called the
geometric sequence with ratio ![$\alpha$](img12.gif)
.
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
![$\alpha\in\mbox{{\bf C}}$](img10.gif)
, then the sequence
![$\displaystyle { g_\alpha\colon
n\mapsto\sum_{j=0}^n\alpha^j}$](img18.gif)
is called the
geometric series with ratio
![$\alpha$](img12.gif)
.
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
![$\{\alpha^n\}$](img21.gif)
and the
figure for
![$\displaystyle {\{\sum_{j=0}^n\alpha^j\}}$](img22.gif)
. Then prove that this similarity is
really present for all
![$\alpha\in\mbox{{\bf C}}\backslash\{1\}$](img23.gif)
.
7.5
Definition (Constant sequence.)
For each
![$\alpha\in\mbox{{\bf C}}$](img10.gif)
, let
![$\tilde\alpha$](img24.gif)
,
denote the constant sequence
![$\tilde\alpha\colon n\mapsto\alpha$](img25.gif)
; i.e.,
![$\tilde\alpha=\{\alpha,\alpha,\alpha,\alpha,\cdots\}$](img26.gif)
.
Next: 7.2 Convergence
Up: 7. Complex Sequences
Previous: 7. Complex Sequences
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