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** 11.1**
**Definition (Series operator.)**
If

is a complex sequence, we define a new sequence

by

or

We use variations, such as

is actually a function that maps complex sequences to complex sequences. We
call

the

*series* corresponding to

.

** 11.2**
**Remark.**
If

are complex sequences and

, then

and

since for all

,

and

** 11.3**
**Examples.**
If

is a geometric sequence,

then

is a
sequence we have been calling a geometric series.

If

, then

is the sequence for

that we
studied in the last chapter.

** 11.4**
**Definition (Summable sequence.)**
A complex sequence

is

*summable*
if and only if the series

is convergent. If

is summable, we denote

by

. We call

the

*sum* of

the series

.

** 11.5**
**Example.**
If

and

, then

.

** 11.6**
**Example (Harmonic series.)**
The series

is called the

*harmonic series*,

and is denoted by

. Thus

We will show that

diverges; i.e., the sequence

is not summable. For all

, we have

From the relation

, we have

and (by induction),

Hence,

is not bounded, and thus

diverges; i.e.,

is not summable.

Proof: The proof is left to you.

** 11.8**
**Exercise.**
Let

be summable sequences. Show that

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

is summable and

.
However,

so

. Thus

is
unbounded and hence

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
** Index**