One example of a sequence that is not monotonic but is convergent is the sequence defined by:
a_n = (-1)^n/n
As n increases, this sequence alternates between positive and negative values, approaching zero. Specifically:
- For n = 1, a_1 = -1
- For n = 2, a_2 = 1/2
- For n = 3, a_3 = -1/3
- For n = 4, a_4 = 1/4
- For n = 5, a_5 = -1/5
- And so on…
Clearly, this sequence does not consistently increase or decrease, hence it is not monotonic. However, as n approaches infinity, the terms of the sequence get closer and closer to 0, indicating that the sequence converges to:
lim (n→infinity) a_n = 0
To address the second part regarding the truth value of the statement, we must know what the statement is. However, the explanation of why a non-monotonic sequence can still converge to a limit is due to the fact that convergence depends on the behavior of the terms as n approaches infinity rather than the order of those terms. Therefore, it is possible for a sequence to oscillate and yet still converge.