11. Infinite series of constants.
Def. An infinite series is an expression that can be written in the form :
+
+ . . . +
+ . . . =
(1)
The numbers , , . . . , . . . are called the terms of the series : is the n-th term ( or general term) of the series.
Let
denote the sum of the initial terms of the series , up to
including the term with index n
.
Thus,
=
,
=
+
,
=
+
+
. . . . . . . . . . . . .
= + + . . . +
The
number
is called the n-th
partial
sum
of the series and the sequence
,
is called the sequence of partial sums.
Def.
Let
be the sequence of partial sums of the series (1).
If
the sequence
converges
to a limit
,
then the series is said to converge to S
,
S is called the sum of the series
= S
If
the sequence
of partial sums diverges (
=
or
),
then the series is said to diverge.
12. Harmonic series.
Is one of the most important of all diverging series.
=
1+
+
+
+ . . . +
+ . . .
Proof: The terms in the series are all positive , the partial sums:
= 1
= 1+
= 1+ +
From a strictly increasing sequence
…
Lemma
.
If a sequence
is eventually increasing , then there are two possible
( a ) There is a constant M called an upper bound for the sequence , converges to a limit L satisfying L M .
(
b )
No upper bound exist in which case
= +
.
Thus , by Lemma we can prove divergence that there is no constant M that is greater then or equal to every partial sum.
To this end , we will consider selected partial sums , namely ,
,
,
,
,
. . .
Note that the subscripts are successive powers L , so that there are partial sims satisfying the unequalities :
=
1+
=
+
=
=
+
+
+
+
=
+
=
+
+
+
+
+
+
+
+
=
+
=
+ +
If
M
is any constant we can find a positive integer
n
such that
M.
But
for this n
M
so
that no constant M is greater than or equal to every partial sum of
the harmonis series.
This proves diverges.
13.Cauchy’s convergence criterion.
∞
A series ∑ An converges iff for every Ԑ >0 there is nԐ<N and any p€ N
n=1
|Sn+p - Sn|<Ԑ |an+1 +an+2 +..+an+p|<Ԑ (1) ∞
Proof: In terms of the partial sums {Sn} at ∑ an, Sn+p – Sn= an+1 +an+2 +..+an+p
n=1
Therefore, (1) can be written as |Sn+p - Sn|<Ԑ. Since ∑ an converges if and only if {Sn} converges. 13.Cauchy’s convergence criterion for sequence implies the conclusion.