Alternating series
We refer to a series as an alternating series if its terms' signs
alternate, that is, the sequence looks like this

  1 an  a1  a2  a3 
n
n 1
where every an is positive.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
The convergence of an alternating series is governed by a
theorem proved by Leibnitz
If an n 1 is a decreasing sequence of positive numbers

which converges to zero, then the alternating series

  1
n 1
n
an  a1  a2  a3 
converges and its sum s satisfies the inequalities
a1  a2  s  a1
Examples
1 1 1
The series 1    
2 3 4
converges
In fact, we even can calculate the sum of this series, which is ln 2

So does the series

n 1
 1
n
a
n
where a > 0
Absolutely convergent series

Let
a
n
n 1

be a series. We say that
a
n 1
n
is absolutely
convergent or that it absolutely converges if the series

a
n 1
converges
An absolutely convergent series is convergent
n
Relatively convergent series

Let
a
n 1
n

be a series. We say that
a
n 1
n
is relatively convergent

or that it relatively converges if it converges, but the series  an
n 1
diverges
Examples
1 1 1
The series 1    
2 3 4
converges relatively since it
converges due to the Leibnitz theorem, but the series

1
1 1 1

1

  

2 3 4
n 1 n
which is known as the harmonic series, diverges (as can be
established, for example, by the integral criterion).

Let
a
n 1
n
be a relatively convergent series. Put
 0 for an  0
pn  
an for an  0
 0 for an  0
qn  
an for an  0
qn  
Then lim  pn   and lim

n 
n 
lim pn  0
n 
and
lim qn  0
n 
Proof

a
converges and so lim an  0 and thus also lim an  0
n
n 1
n 
n 
Since 0  pn  an and 0  qn  an , this proves that
lim qn  0 and lim qn  0
n 
n 
p
If
n
 p
n
and
q
n
both were convergent, then also their sum
 qn    an
would be convergent.

 

lim  pn  p and lim  qn    lim  an  lim   pn   qn   
n 
n 
n 
n
n 1
n 1
n 1
n 1
 n 1





 

lim  pn   and lim  qn  a  lim  an  lim   pn   qn   
n 
n 
n 
n 
n 1
n 1
n 1
n 1
 n 1




Rearranging the terms of a series

Let us have two series
a
n 1
n

and
a
in
n 1


We say that
a
n 1
in
has been formed from
n 1

rearrangement or is a rearranged
a
n 1
a
n
n
by a term
if in the sequence
n1 , n2 , n3 ,
each natural number occurs exactly once
Example
a1  a2  a3  a4  a5  a6  a7  a8  a9  a10  a11  a12  a13 
a2  a1  a4  a3  a6  a5  a8  a7  a10  a9  a12  a11  a14 

Let
a
n 1
n
absoltely converge. Then no rearrangement of its terms
can change its sum.
The following remarkable theorem is due to Riemann:

Let
a
n 1
n
be a relatively convergent series and let S be an
arbitrary real number or S   . Then the terms of

a
n 1
can be rearranged so that

lim  an  S
n 
n 1
n
Example instead of proof
1 1 1 1 1 1 1 1 1 1 1 1 1 1
1             
2 3 4 5 6 7 8 9 10 11 12 13 14 15
S  1.5
1 1
1    1.53333  1.5
3 5
1 1 1
1     1.03333  1.5
3 5 2
1 1 1 1 1 1 1 1
1          1.52180  1.5
3 5 2 7 9 11 13 15
1 1 1 1 1 1 1 1 1
1           1.2718  1.5
3 5 2 7 9 11 13 15 4
1 1 1 1 1 1 1 1 1 1 1 1 1
1
1             
 1.51435  1.5
3 5 2 7 9 11 13 15 4 17 19 21 23 25
1.5