# MTH 251 Lecture Notes - Lecture 26: Alternating Series Test, Conditional Convergence, Direct Comparison Test

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MTH 251-Lecture 26-The ratio and root tests
Ratio Test Proof:


There is an N, so that if n, 

For



Generally,
 for

This series is geometric and converges, so,  
  
 converges, so
 
converges.
Example:
Does 
 converge or diverge?
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Use the ratio test…









This is indeterminate, so, bring down the power,







So, 

=
, so
 is absolutely convergent by the ratio test.
Recall, however that it would be much quicker to show convergence by the comparison test.
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