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,  a1a2a3, . . . , an, . . .  all of the terms of which are added together, where an 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
 second and third
 Therefore,
since the parentheses equals already known value s3.
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  s3 + 9 + 27 > 3.
 Generally, it can be written that is, the partial sums sn increase to infinity as  n ® oo.
Therefore, the harmonic series diverges.
Functions contents E