Name__________________________________
Calculus
Period ________
Lesson 9.5: Alternating Series, Absolute Convergence and Conditional Convergence
Theorem 8: The Alternating Series Test (Leibniz’s Theorem)

The series
  1
n 1
n 1
un  u1  u2  u3  u4  ... converges if all three of the following are satisfied.
1. The un ' s are all positive.
2. un  un 1 for all n
3. lim un  0
n
Absolute Convergence
A series  an converges absolutely if the corresponding series of absolute values  an converges.
A series that converges, but does not converge absolutely, converges conditionally.
Determine whether the series converges absolutely, converges conditionally or diverges. Give reasons for your
answer. Find a bound for the truncation error after 99 terms.
1 1 1
1 1
1. 1     ...
2. 1  2  3  4  ...
3. 2  1    ...
2 3 4
2 4

4.
  1
n 1
n 1
 1 


 n

5.
  1
n 1
n 1
1
n2
Radius of Convergence
To determine to what values each power series converge, we use one of three tests: Geometric, Root, or Ratio.
When using the Root or Ratio test, you must check the endpoints individually because the test is inconclusive at
 1.
In Exercises 35-50, find (a) the interval of convergence of the series. For what values of x does the series
converge (b) absolutely, (c) conditionally. Also, identify the Radius of Convergence.

36.
  x  5
n
n 0

38.

n 0
 3x  2 
n
n

44.
nx n

n
2
n  0 4  n  1
HW p523: 23-25, 35-43 odd