Series
Alternating series
Alternating series test or Leibnitz's alternating series test
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 s2n 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.
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