Determine whether the sequence converges or diverges. If it converges, find the limit. (If an answer does not exist, enter DNE.)
{0, 6, 0, 0, 6, 0, 0, 0, 6, ...}
read up on the definition of convergence.
You will see that this sequence does not converge.
To determine whether the given sequence converges or diverges, we need to identify a pattern in the terms.
The given sequence is {0, 6, 0, 0, 6, 0, 0, 0, 6, ...}.
Looking closely, we can observe that the sequence alternates between the terms 0 and 6. Specifically, we have three consecutive 0's followed by a single 6. This pattern repeats indefinitely.
Since there is no specific value to which the terms of the sequence are approaching, we can conclude that the sequence diverges.
Therefore, the answer is that the given sequence diverges.
To determine whether the given sequence {0, 6, 0, 0, 6, 0, 0, 0, 6, ...} converges or diverges, we need to observe the pattern and behavior of the terms in the sequence.
Looking at the sequence, it appears to repeat a pattern consisting of two consecutive zeros and one 6. This pattern then repeats indefinitely.
One way to formally prove this is to observe that the position of a 6 in the sequence can be represented by the equation n = 3k + 2, where n is the term number and k is a non-negative integer. Similarly, the positions of the zeros can be represented by either n = 3k or n = 3k + 1.
Since the sequence follows this repeating pattern, we can conclude that it is a periodic sequence. Moreover, we can see that the positive terms (6) keep repeating, while the zero terms alternate.
For a converging sequence, all the terms should approach a certain value as we move further along the sequence. However, in this case, the sequence does not approach a single number but instead oscillates between two values, 0 and 6.
Therefore, we can say that the given sequence diverges, which means it does not have a finite limit.