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MAT136H1 Study Guide - Midterm Guide: Ratio Test, Conditional Convergence

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11.6 Infinite Sequences & Series
Absolute Convergence & Ratio and Root Tests
Question #2 (Medium): Convergence by Ratio Test
By the word “ratio” test, it takes and the next term expression  and takes a ratio of the two so
1) 
 is absolutely convergent (ie. convergent)
2) 
, or means
 is divergent
3) 
means no conclusion can be drawn from the Ratio Test about convergence or
divergence of
All series expression falls into one of the three categories, thus depending on the outcome of the ratio
test, its convergence, or divergence, or no conclusion can be drawn based on the test.
Sample Question
Determine if the series is absolutely convergent conditionally convergent, or divergent.
To put into ratio test, is just taking the series expression inside the summation as it is, thus 
The ratio text requires the next term expression , thus it is then  
 simply replacing
by wherever occurs.
Now that these two expressions are written down, the ratio can be taken:
 then it can be simplified so that limit value
becomes more obvious: 
 
Since this is less than 1, the series is absolutely convergent, thus the series 
 is convergent.
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