Series
Tests for convergence
Ratio test
Root test or Cauchy's root test
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:  Examine whether the series converges or diverges.
 Solution:  Since,
the series converges if  0 < x < 1.
 Example:  Determine whether the series converges or diverges.
 Solution:  Since,
therefore L < 1 and the series converges.
If the given series has a factorial in the nth term expression, the ratio test should be used.
 Example:  Let show that the logarithmic series converges.
 Solution:
 thus, the series converges for x < 1. The sum of the series
Root test or Cauchy's root test
 The test for whether a series of positive terms is absolutely convergent by considering;
 (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:  Determine whether the series converges or diverges.
 Solution:  The given geometric series converges since
 Example:  Let show that the series converges.
 Solution:  The series converges since
 Example:  Test the series for convergence.
 Solution:  The series converges since
We use the root test when the entire nth term expression is raised to some power of n.
Calculus contents B
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