Series
      Infinite series
         The remainder or tail of the series
         Necessary and sufficient condition for the convergence of a series
         Necessary condition for the convergence of a series
         The n-th term test for divergence of series
      Properties of series
         The product of two series or the Cauchy product
      Geometric series
      P-series
The remainder or tail of the series
As with sequences, the convergence of an infinite series  only depends on the behavior of the
general term of the series an as n increases to infinity, and not on any finite number of its initial terms.
Note that, since  
the series converges if and only if converges.
Therefore, to show that a series converges we can ignore any finite number of terms  at the
beginning, and just need to prove the convergence of the tail or remainder of the series.
The difference between the sum s of a convergent series a1 + a2 + a3 + . . . + an + . . .  and the nth partial sum sn is called the remainder (tail) rn of the series, i.e.,
rn = s - sn  = an + 1 + an + 2 + an + 3 + . . .    or     s = sn + rn.
Thus, if a series converges then the remainder rn  = an + 1 + an + 2 + an + 3 + . . .  converges too,
that is, since   and  s  = sn + rn  then,   
Necessary and sufficient condition for the convergence of a series - Cauchy's convergence test
Necessary and sufficient condition that the sequence of partial sums {sn} of a given series converges, and 
hence the series converges, is that for given however small positive number e it is possible to find
an index n0 such that   | sn + r - sn | < e  whenever  n > n0 (e) and  r = 1, 2, 3, . . . ,
or expressed by terms of the series, if
| an + 1 + an + 2 + an + 3 + . . . + an + r  | < e  whenever  n > n0 (e) and  r  = 1, 2, 3, . . . 
Therefore, a series converges if the absolute value of the sum of any finite number of sequential terms can become arbitrary small by starting the addition from a term which is far enough.
Necessary condition for the convergence of a series
Hence, it is necessary condition for the convergence of a series that its terms tend to zero as n
increases to infinity, that is   So, if this condition is not satisfied the series diverges.
That this condition is only necessary but not sufficient condition for the convergence shows the harmonic series for which
as was shown in previous section.
Necessary condition for the convergence of a series is usually used to show that a series does not converge.
The nth term test for divergence
Note that this is only a test for divergence. That is, if we can prove that the sequence {an} does not
converge to 0, then the infinite series does not converge.
Properties of series
If given are two convergent series,
then convergent is the series obtained by adding or subtracting their same index terms, and its sum equals the sum or the difference of their individual sums, i.e.,
The product of two series or the Cauchy product
If given are two convergent series of positive terms, then the product
denotes the convergent series sum of which is equal to the product of the sums of the given series.
Geometric series
A series, whose successive terms differ by a constant multiplier, is called a geometric series and written as
If  | x | < 1 ;    the nth partial sum is
    Thus, the geometric series is convergent if  | x | < 1 and its sum is 
If  | x | > then  the geometric series diverges.
Example:  Show that the series   converges.
Solution:   Given is the geometric series subsequent terms of which are multiplied by the factor 1/2.
Example:  Let prove that the pure recurring decimal  0.333 ... converges to 1/3.
Solution:   Given decimal can be written as
Example:  Let calculate the square of the convergent geometric series
using the Cauchy product shown above.
Solution:
                                   = 1 + 2x + 3x2 + 4x3 + + (n + 1)xn +  
the obtained series converges for 0 < x <1.  
P-series
The series converges if  fixed constant  p > 1 and diverges if  p < 1.
By grouping terms 
where the number of terms in parentheses form the sequence  2, 4, 8, ... 2r-1, ... such that
therefore, each value in parentheses is smaller than the corresponding term of the geometric series
Thus, if   p > 1 then  q < 1,  the geometric series converges so that the given series is also convergent.
Euler discovered and revealed sums of the series for  p = 2m, so for example
If   p < 1  then  np < n  or 1/np > 1/n, therefore the terms of the given series are not smaller than the terms
of the divergent harmonic series so, given series diverges.
Functions contents E
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