Question

Use the definition of the limit of a sequence to show that ((-1)ⁿ (1 + 1/n)) is a divergent sequence.

Answer 1

To show that the sequence ((-1)n (1 + 1/n)) is divergent, we need to show that it does not converge to a specific limit as n approaches infinity.

Step-by-step explanation:

To show that the sequence ((-1)n (1 + 1/n)) is divergent, we need to show that it does not converge to a specific limit as n approaches infinity.

Let's assume that the sequence converges to a limit L. Then, for any given epsilon (ε) greater than 0, we should be able to find an integer N such that for all n > N, |((-1)n (1 + 1/n)) - L| < ε.

However, by considering the two subsequences (1 + 1/n) and (-1)n, we can show that the absolute difference |((-1)n (1 + 1/n)) - L| can never be smaller than a certain positive value, regardless of the chosen N and epsilon. Therefore, the sequence is divergent.