11.6 Infinite Sequences & Series
Absolute Convergence & Ratio and Root Tests
Question #3 (Medium): Alternating Series Tested Using the Ratio Test
The ratio test states:
1) | | means ∑ 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 ∑
Notice that the ratio is inside the absolute value sign. Therefore, even alternating series can be
tested for its convergence or divergence using this ratio test.
Determine if the series is absolutely convergent, conditionally convergent, or divergent.
∑ ( ) ( )
The given series is an alternating series, however it can still be tested against the ratio test to see any
conclusion can be drawn about its convergence, divergence.
Since the ratio is placed inside the absolute value sign for the ratio test, the alternating sign