

Series 
Infinite
series 
Harmonic
series 





Infinite series 
An
infinite series is the sum of infinite sequence of terms which we
denote 

That
is, given an infinite sequence of real numbers, a_{1},
a_{2}, a_{3}, . . . , a_{n},
. . .
all of the terms of which are added
together, where a_{n}
denotes the general term of the series. 

Harmonic
series 
The
series of the reciprocals of natural numbers 

is
called the harmonic series since the middle term of any three successive
terms is the harmonic mean of the
other two. 
Given
three positive numbers, a,
b
and c
are said to be in the harmonic proportion if 

that
is, if b
is the harmonic mean of the numbers, a
and c. 
Therefore,
any three successive terms, 

of
the harmonic series, are in the harmonic 

proportion since 


As,
in the harmonic proportion a
> c then, a

b > b 
c therefore a
+ b > 3b 
c 
or
a
+ b + c > 3b. 
That
is, the sum of any three successive terms of the harmonic series is
three times greater than the middle term.
P. Mengoli (1626  1686) used this property to prove divergence of the
harmonic series. 
Let
examine behavior of the sequence of partial sums of the harmonic series
applying this property. 
Thus,
the sum of the three successive terms beginning from the second gives 

Next
nine terms, from n = 5 to
13, divide in three groups with three terms in
each, so that; first gives 

Therefore, 



since
the parentheses
equals already known value s_{3}. 
For
next 27
terms, from n =
14 to 40,
following the same procedure, we get that their sum is greater
than 


for
which is already shown that is >
1,
thus s_{3 +
9 + 27
}> 3. 


Generally,
it can be written 

that
is, the partial sums s_{n} increase
to infinity as n
®
oo. 

Therefore,
the harmonic series diverges. 








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