Question

Give an example of a bounded sequence without a limit.

Short answer

Question: Provide an example of a sequence that is bounded but does not converge to any limit. Answer: An example of such a sequence is a_n = (-1)^n, where n is a natural number. This sequence is bounded between -1 and 1 but does not converge to any limit as it oscillates between the two values indefinitely.

Step-by-step solution

Example of a bounded sequence without a limit

One such example is the sequence (-1)^n, where n is a natural number. Step by step breakdown of the sequence:

Define the sequence

Define a sequence a_n = (-1)^n, where n is a natural number (n = 1, 2, 3, ...).

Analyze the sequence pattern

Observe how the terms in the sequence alternate as the value of n increases: a_1 = (-1)^1 = -1, a_2 = (-1)^2 = 1, a_3 = (-1)^3 = -1, a_4 = (-1)^4 = 1, and so on.

Determine if the sequence is bounded

Since the sequence alternates between -1 and 1, it is clear that the sequence is bounded. We can express this by saying: -1 ≤ a_n ≤ 1, for all natural numbers n.

Determine if the sequence has a limit

The sequence does not converge to a single value, as it continues to oscillate between -1 and 1 indefinitely. Therefore, the sequence does not have a limit.