Calculus
Lesson
52
Alternating
Series
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Back to Dr. Nandor's Calculus Page
So far, we've only looked at series where each term was +ve, except for
simple geometric series.
An example of an alternating series:
Alternating series test:
converges if
1) 
for all n (take
derivative!!!!)
and 2) 
note 
Example: 
converges
even though the harmonic doesn't.
Alternating series remainder:
IF S exists. Proof on page 583, but it's really
common sense from the conditions above!
Example: 
Approximate
S using 6 terms
Use the alternating series test to check...it does converge.
Using the first 6 terms, 
So
Absolute vs. Conditional Convergence
(NOT JUST FOR PURE ALTERNATING SERIES!!!!)
if 
converges, then 
(note the converse is NOT true! ex/harmonic)
if 
converges, then 
is absolutely
convergent
if 
converges but 
is divergent,
then 
is conditionally
convergent.
Examples: 
abs.
div.
cond.
cond.
Note that rearrangement doesn't work for alternating series!!!!!!
