Determine whether the sequence converges or diverges. If it converges, find the limit. (If an answer does not exist, enter DNE.)
{0, 8, 0, 0, 8, 0, 0, 0, 8, ...}
lim n→∞ an =
If you have a few million zeros and one eight, I would be inclined to say it converges to zero.
to be convergent, the sequence must approach a limit.
Clearly, 0 is not the limit, as you can always find another term which is 8.
the limit is L if for every n>N, |a_n - L| < ε, for any ε>0 for some N.
To determine if the sequence converges or diverges, we first observe the pattern of the terms. The sequence provided is {0, 8, 0, 0, 8, 0, 0, 0, 8, ...}.
From the pattern, we can see that the terms alternate between 0 and 8. Specifically, the terms are 0 for every position that is not a multiple of 3, and 8 for every position that is a multiple of 3.
Since the sequence does not approach a specific value and the terms oscillate between two different values (0 and 8), the sequence does not converge.
Therefore, the limit does not exist (DNE).