Series
      Infinite series
         The sequence of partial sums
         The sum of the series
         Convergence of infinite series
         Divergence of infinite series
         Convergent and divergent series examples
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.
The sequence of partial sums
Recall that we've already dealt with infinite series in previous section when representing real numbers as infinite decimals. Thus, for example the real number
so, when adding up this infinite number of terms, we start by calculating its partial sums
Thus, obtained is sequence  {sn} of the partial sums,  s1s2s3, . . . ,  sn, . . .  where sn denotes the sum of the first n terms of the series and is called the nth partial sum.
Therefore, the limit of the sequence {sn} as n tends to infinity, i.e.,
equals the value of the real number 0.333 . . .  that is, equals the sum of the infinite series.
The sum of the series
An infinite series has a sum if the sequence of its partial sums converge to a finite number s, i.e., 
The sum of an infinite series is the limit of its partial sums as n tends to infinity.
Let rewrite the above example using this notation,
where,
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.
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