Series
      Infinite series
      Alternating series
         Alternating series test or Leibnitz's alternating series test
         Absolute convergence
         Conditional convergence
      Series of positive terms
Alternating series
A series the terms of which are alternately positive and negative is called the alternating series.
Alternating series test or Leibnitz's alternating series test
An alternating series converge if the absolute values of its terms decrease monotonically to zero as n tends to infinity.
Given an alternating series    a1 - a2 + a3 -  + + (-1)n - 1 an +  
suppose      a1 > a2 > a3 > > an > an + 1 > > 0   and  
The sequence of the partial sums sn of the series alternate approaching the same limit s. The terms of s2 of the even indices, increase while the terms of s2n + 1 of the odd indices, decrease as is shown on the number line below. Every term of the sequence of an even index is smaller than every term of an odd index therefore,   s2n < s < s2n + 1
and  
Example:  Let show that the series   converges.
Solution:  The given alternating series converges since the sequence of absolute values of its terms 
  that is,  
decrease to zero as n tends to infinity, or  The series converges to  ln 2  or  s = ln 2.
Note that the given logarithmic series is not absolutely convergent since
obtained harmonic series diverges.
Therefore, the series is said to be conditionally convergent.
Example:  Let show that the series   converges.
Solution:  The given alternating series converges since the sequence of absolute values of its terms 
  that is,  
decrease to zero as n tends to infinity, or  As Leibnitz proved, the series converges to p/4.
Absolute convergence
An infinite series is called absolutely convergent if the infinite series is convergent, i.e.,
the series of absolute values of its terms converge.
Thus, if a series is absolutely convergent, then both series,
Example:  Let show that the series   is absolutely convergent.
Solution:  Given series is absolutely convergent since the series of absolute values of its terms
  converges to 2,  
Therefore, the series is convergent and its sum is 2/3.
Example:  Let show that the series   is absolutely convergent.
Solution:  Given series is absolutely convergent since the series of absolute values of its terms
Therefore, the series converges and its sum
Conditional convergence
If an infinite series is not convergent, it is called divergent. However, if it is convergent but not
absolutely convergent, it is called conditionally convergent.
Series of positive terms
Suppose the series consists of positive terms only. Then its partial sums form an
increasing sequence, therefore given series converges if and only if the sequence of partial sums {sn} is bounded.
We use this property for convergence tests of series with positive terms.
Example:  The series of positive terms
converges since
that is, the sequence of partial sums is bounded above, and the series converges to the number e.
If we calculate the partial sum of first 10 terms of the series, s10 = 2.718281 obtained is the number e to six correct decimal digits what shows that the series increases relatively fast.
Calculus contents B
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