Series
     Infinite series
      Geometric series
      P-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.
Calculus contents B
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