Series
      Infinite series
         Convergence of infinite series
         Divergence of infinite series
         Convergent and divergent series examples
      Harmonic series
Convergence of infinite series
If the limit of the sequence of partial sums exists as a real number, then the series is convergent.
Divergence of infinite series
If the limit of the sequence of partial sums does not exist or tends to  + oo  or  - oo
then, the series is divergent or oscillates.
Convergent and divergent series examples
Example:  Show that the series   diverges.
Solution:   The given infinite sum of natural numbers is called the arithmetic series.
Using the formula for the sum of the arithmetic sequence, whose difference d = 1, we calculate the sum of the first n terms of the series.
Example:  Let show that the series converges.
Solution:   We use the method of partial fraction decomposition to rearrange given expression.
Since the denominator consists of linear factors then, we can write
 Therefore,
Since the second and the first term of any two successive parentheses cancel then, the sum of the first n terms of the series
therefore the series converge, and its sum  s = 1.
Example:  Show that the series   converges.
Solution:   Use the method of partial fraction decomposition to rearrange given expression.
Since the number of the coefficients used in the expansion relates to the degree of the polynomial in the denominator, and as it consists of linear factors some of which are repeated then,
 Therefore,  
Since the second and the first term of any two successive parentheses cancel then, the sum of the first n terms of the series
therefore the series converge, and its sum  s = 1.
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.
Calculus contents B
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