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11.1
Definition (Series operator.)
If
![$f$](img13.gif)
is a complex sequence, we define a new sequence
![$\sum f$](img1335.gif)
by
or
We use variations, such as
![$\sum$](img1339.gif)
is actually a function that maps complex sequences to complex sequences. We
call
![$\sum f$](img1335.gif)
the
series corresponding to
![$f$](img13.gif)
.
11.2
Remark.
If
![$f,g$](img109.gif)
are complex sequences and
![$c\in\mbox{{\bf C}}$](img1067.gif)
, then
and
since for all
![$n\in\mbox{{\bf N}}$](img9.gif)
,
and
11.3
Examples.
If
![$\{r^n\}$](img1344.gif)
is a geometric sequence,
then
![$\sum\{r^n\}=\{\sum_{j=0}^nr^j\}$](img1345.gif)
is a
sequence we have been calling a geometric series.
If
![$\displaystyle {\{c_n(t)\}=\left\{{{t^{2n}(-1)^n}\over {(2n)!}}\right\}}$](img1346.gif)
, then
![$\sum\{c_n(t)\}=\{C_n(t)\}$](img1347.gif)
is the sequence for
![$\cos(t)$](img1348.gif)
that we
studied in the last chapter.
11.4
Definition (Summable sequence.)
A complex sequence
![$\{a_n\}$](img391.gif)
is
summable
if and only if the series
![$\sum\{a_n\}$](img1349.gif)
is convergent. If
![$\{a_n\}$](img391.gif)
is summable, we denote
![$\lim(\sum\{a_n\})$](img1350.gif)
by
![$\displaystyle {\sum_{n=0}^\infty a_n}$](img1351.gif)
. We call
![$\displaystyle {\sum_{n=0}^\infty a_n}$](img1351.gif)
the
sum of
the series
![$\sum\{a_n\}$](img1349.gif)
.
11.5
Example.
If
![$r\in\mbox{{\bf C}}$](img1353.gif)
and
![$\vert r\vert<1$](img1354.gif)
, then
![$\displaystyle {\sum_{n=0}^\infty r^n=\lim\{\sum_{j=0}^n
r^j\}={1\over {1-r}}}$](img1355.gif)
.
11.6
Example (Harmonic series.)
The series
is called the
harmonic series,
and is denoted by
![$\{H_n\}_{n\geq 1}$](img1357.gif)
. Thus
We will show that
![$\{H_n\}_{n\geq 1}$](img1357.gif)
diverges; i.e., the sequence
![$\displaystyle { \left\{ {1\over n}\right\}_{n\geq 1}}$](img118.gif)
is not summable. For all
![$n\geq 1$](img1360.gif)
, we have
From the relation
![$\displaystyle {H_{2n}\geq{1\over 2}+H_n}$](img1362.gif)
, we have
and (by induction),
Hence,
![$\{H_n\}_{n\geq 1}$](img1357.gif)
is not bounded, and thus
![$\{H_n\}$](img1359.gif)
diverges; i.e.,
![$\displaystyle { \left\{ {1\over n}\right\}_{n\geq 1}}$](img118.gif)
is not summable.
Proof: The proof is left to you.
11.8
Exercise.
Let
![$f,g$](img109.gif)
be summable sequences. Show that
![$f+g$](img169.gif)
is summable and that
11.9
Example.
The product of two summable sequences is not necessarily summable. If
then
This is a null sequence, so
![$f$](img13.gif)
is summable and
![$\displaystyle {\sum_{n=1}^\infty f(n)=0}$](img1370.gif)
.
However,
so
![$\displaystyle {\left(\sum(f^2)\right)(2n)=2\sum_{j=1}^n{1\over j}=2H_n}$](img1372.gif)
. Thus
![$\sum(f^2)$](img1373.gif)
is
unbounded and hence
![$f^2$](img1374.gif)
is not summable.
11.10
Theorem.
Every summable sequence is a null sequence. [The converse is not true. The harmonic
series provides a counterexample.]
Proof: Let
be a summable sequence. Then
converges to
a limit
, and by the translation theorem
also.
Hence
i.e.,
and it follows that
is a null sequence.
Next: 11.2 Convergence Tests
Up: 11. Infinite Series
Previous: 11. Infinite Series
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