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

(1/3,1,1/4,1/2,1/5,1/3,1/6,1/4...)

The sequence can be rearranged as

1/3, 1/4, 1/5, 1/6, ...
....1, 1/2, 1/3, 1/4, ...
so, the terms converge to zero

The sum, however, will come close to twice the sum of the harmonic series, which diverges.

Well, this sequence seems to be jumping around quite a bit! It's like a roller coaster of numbers.

Let's take a closer look at the pattern: (1/3, 1, 1/4, 1/2, 1/5, 1/3, 1/6, 1/4...)

Hmm... it doesn't seem to be settling down to a particular value. It keeps changing and bouncing around, so it's like trying to catch a slippery fish with your bare hands.

Therefore, this sequence diverges, my friend. There is no limit to catch here. It's like trying to catch a unicorn or find a pot of gold at the end of a rainbow. Impossible!

So, in short, the sequence does not converge, and the limit is DNE – just like trying to catch a clown with a fishing net. Good luck with that!

To determine whether the given sequence converges or diverges, we need to examine the pattern of the terms.

From the given sequence: (1/3, 1, 1/4, 1/2, 1/5, 1/3, 1/6, 1/4...)

We can observe that the terms are alternating between fractions with the denominator decreasing by 1 in each step.

Let's break the sequence into two subsequences:
Subsequence A: (1/3, 1/4, 1/5, 1/6, ...)
Subsequence B: (1, 1/2, 1/3, 1/4, ...)

Now, let's consider the limits of each subsequence separately:

For subsequence A:
As the denominator approaches infinity, the terms of subsequence A approach 0. So, the limit as n approaches infinity for subsequence A is 0.

For subsequence B:
The terms of subsequence B are just the reciprocals of the positive integers, which can be written as 1/n. As n approaches infinity, 1/n approaches 0. So, the limit as n approaches infinity for subsequence B is 0 as well.

Since both subsequences have a limit of 0, we can conclude that the given sequence converges and the limit is 0.

To determine whether the sequence converges or diverges, we need to analyze its behavior as the terms progress.

Looking at the sequence:

(1/3, 1, 1/4, 1/2, 1/5, 1/3, 1/6, 1/4...)

We can see that the terms alternate between fractions with denominators that keep decreasing, then start over. The repeating pattern suggests that the sequence may not converge to a single limit.

To find out if the sequence converges or diverges, we can consider the subsequences formed by only selecting terms with the same denominator. Let's examine them one by one.

Subsequence 1: (1/3, 1/3, 1/3, ...)
This subsequence clearly converges to the value 1/3.

Subsequence 2: (1/4, 1/4, 1/4, ...)
This subsequence also converges to the value 1/4.

Subsequence 3: (1/5, 1/5, 1/5, ...)
Like the previous subsequences, this subsequence converges to the value 1/5.

Therefore, we can observe that each subsequence formed by selecting terms with the same denominator converges to a specific value. However, since the subsequences correspond to different denominators, we cannot identify a single limit for the entire sequence. Hence, the sequence diverges.

In conclusion, the sequence (1/3, 1, 1/4, 1/2, 1/5, 1/3, 1/6, 1/4...) diverges and does not have a limit.