Determine whether the sequence converges or diverges. If it converges, find the limit. (If an answer does not exist, enter DNE.)

{0, 4, 0, 0, 4, 0, 0, 0, 4, ...}

Oh, this sequence is doing a little dance, going back and forth between 0 and 4. It's like a fickle pendulum swinging from one extreme to the other. So, does it converge? Well, nope! It's going back and forth, never settling down to a single value. So, we can say it diverges. It's a sequence that just can't make up its mind!

The given sequence {0, 4, 0, 0, 4, 0, 0, 0, 4, ...} appears to have a repeating pattern where the number 4 is repeated every three terms, and the other terms are all zeros.

To determine whether the sequence converges or diverges, we can find the limit of the sequence.

Let's denote the terms of the sequence as a_n, where n is the index of the term.

We can observe that for every n that is divisible by 3, a_n = 4. For all other values of n, a_n = 0.

Therefore, the limit as n approaches infinity, lim(a_n) is equal to 0, since the terms of the sequence will eventually become zeros and stay zeros for all subsequent terms.

Hence, the sequence converges to the limit of 0.

To determine whether the given sequence converges or diverges, we need to analyze the pattern and behavior of its terms.

Looking at the given sequence {0, 4, 0, 0, 4, 0, 0, 0, 4, ...}, we observe that the terms alternate between 0 and 4. Specifically, the sequence repeats the pattern {0, 4, 0} indefinitely.

Since the sequence repeatedly cycles between the two distinct terms 0 and 4, it does not approach a specific value as we go further along the sequence. Therefore, we can conclude that the sequence diverges.

In this case, the sequence does not converge, so we cannot find a limit. Hence, the answer is DNE (Does Not Exist).

2+3+4+... = n(n+1)/2 - 1

So, if N = n(n+1)/2 - 1, you know that

the Nth term is 4, and the n+1st term is 0

So, while the number of zeros is infinitely many more than the number of 4's, the terms do not converge to either 4 or 0.