What is an example of a convergent alternating series where the conditions of the alternating series test do not hold?
try this
https://math.stackexchange.com/questions/836801/is-there-a-convergent-alternating-series-that-fails-the-ast
To find an example of a convergent alternating series where the conditions of the alternating series test do not hold, let's first review the conditions of the test.
The alternating series test states that if an alternating series, ∑(-1)^(n-1)*a_n, satisfies the following two conditions, then it converges:
1. The terms (a_n) are positive and decreasing.
2. The limit of the terms, lim(n→∞) a_n = 0.
Now, let's consider an example where the conditions of the alternating series test are not satisfied.
Consider the series ∑(-1)^(n-1)/n. Let's examine the conditions of the alternating series test:
1. The terms of the series are positive since (-1)^(n-1) alternates between -1 and 1, and n is always positive.
2. However, the sequence (1/n) does not satisfy the second condition - the limit of (1/n) as n approaches infinity is not equal to 0, but rather approaches 0.
Therefore, the series ∑(-1)^(n-1)/n is an example of a convergent alternating series where the conditions of the alternating series test do not hold.