Section 8.6: Alternating Series

1. Section 8.6: Alternating Series -

2. Def: In an alternating series terms alternate signs or with δk >0 Then converges Alternating Series test: If is alternating and Is a decreasing sequence Eg: Eg: Eg: Converges to ln2 Diverges because of divergence test Converges by alternating series test But it “just get lucky” without the (-1)k+1 it would diverge

3. Def: converges absolutely if converges If converges ,but not absolutely then it converges conditionally Eg: Eg: Converges absolutely but converges conditionally… Converges absolutely because converges by baby comparison test… Better Ratio Test: Recall the Ratio test required but we could just check to absolute convergence…

4. Error Estimates: notice if satisfies the Alternating series test, then it is “1 step forward, half step back” kind of sum So the difference between the partial sum and the true sum is less than the next term! Rearranging terms: Its probably not too surprising that you can rearrange the terms in an absolute convergence series and still get the same sum, but a conditionally convergent sum can be rearranged to sum to anything you want… One term pushes sum up above the limit, the next pushes it below the limit.. Sum Proof Idea: The positive terms and negative terms both diverge separately.. So add up positive terms until you get past L Subtract negative pieces until you get below L Repeat…since terms 0 the “over shooting” 0