Alternating series test
If decreases to with , then the alternating series converges.
Proposition 1.37 (Alternating Series Test)
Let be a decreasing sequence with and . Then converges.
Example
Example 1.38
converges though diverges, by the alternating series test. [In Section 5, we will show that it converges to .]
Proof
Let be the partial sums. Grouping consecutive terms,
so is increasing and is decreasing. Since ,
so both subsequences are bounded and converge by the monotone convergence theorem. Their limits agree because . A sequence whose odd and even subsequences both converge to the same limit converges to that limit, hence .
Related
Stated in
- Proposition 1.37 (Alternating Series Test)§1.4 Series and Convergence Tests
- Example 1.38§1.4 Series and Convergence Tests