Series
      Tests for convergence
         Comparison test
         Limit comparison test
         Ratio test
Tests for convergence
Comparison test
A series of positive terms converges if its terms, starting from the first or one of the further terms, are not greater than terms of the same index of a known convergent series of positive terms.
Suppose that are series of positive terms. Then
  (1)  If is convergent and an < bn  for all n, then is also convergent.  
  (2)  If is divergent and an > bn  for all n, then is also divergent.  
Example:  Let prove that the series   converges.
Solution:   Since  n2 > (n - 1)n  then  In previous section was shown that the series
converges and its sum  s = 1 therefore, by the comparison test, also converges the series  and the
sum of which as Euler revealed (in 1736).
Recall that the harmonic series, i.e., the series of the reciprocals of natural numbers diverges while the series of the reciprocals of squares of natural numbers converges. The reason is, the terms of the reciprocals of squares decrease faster then the terms of the harmonic series.
The limit comparison test
Suppose are series of positive terms. If
then the series are either both convergent or both divergent.
Example:  Let prove that the series   converges.
Solution:
Since both series are of positive terms, the limit is 1 > 0, and the series  bn = 1/n2  is convergent then, by the limit comparison test given series also converges.
The ratio test
The test for whether a series is absolutely convergent by testing the limit of the absolute value of the ratio of successive terms of the series is called the ratio test.
  (1)  If then the series  is absolutely convergent.  
  (2)  If then the series  is divergent.  
  (3)  If  then the test is inconclusive, i.e., gives no information.  
Example:  Let show that the series   converges.
Solution:  Since, the series converges.
Example:  Let show that the series   converges.
Solution: 
thus, the series converges.
Calculus contents B
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