MATH-M 212 Lecture Notes - Lecture 15: Conditional Convergence, Alternating Series
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Does converge: yes- geometric with, re-write and expand: Theorem 9. 14- alternating series test: let, alternating series and converge if both of these conditions are met: (if limit , series diverges) Do not confuse with term test (nonincreasing) for all. Going back to , use the alternating series test to show its convergence: limit, comparison: For all , it is known that: converges by alternating series test. What happens to the harmonic series when it alternates? o o: for all, converges by alternating series test. Use the first 6 terms to approximate . o o o o: absolute sum of first 6 terms is between and. Definition of absolute and conditional convergence: is same thing as non-alternating version of a series; absolute value makes all terms of positive. If converges, then must also converge: is absolutely convergent if and both converge, is conditionally convergent if converges but diverges.
