Question

Consider the sequence 2/4, 3/5, 4/6, 5/7,... Which statement describes the sequence?

The sequence diverges.

The sequence converges to 0.

The sequence converges to 1.

The sequence converges to [infinity].

Answer 1

The sequence 2/4, 3/5, 4/6, 5/7,... converges to 1. It can be represented as (n+1)/(n+2), and as 'n' becomes very large, the ratio approaches 1 because the terms 1/n and 2/n in the simplified form go to zero.

Step-by-step explanation:

The question you've asked is about a mathematical sequence. Specifically, you want to know if the sequence 2/4, 3/5, 4/6, 5/7,... converges or diverges and to what value it might converge. Let's analyze the pattern. The sequence can be written as (n+1)/(n+2) where 'n' starts at 1 and increases by 1 each time. As 'n' gets very large, the numerator and denominator both approach infinity, but they do so in such a way that their ratio approaches 1.

This is because if we divide both the numerator and the denominator by 'n', we get (1+1/n)/(1+2/n), and as n approaches infinity, both 1/n and 2/n approach 0, leaving us with 1/1, which is 1. Therefore, the sequence does not diverge, nor does it converge to 0 or infinity. Instead, the sequence converges to 1.

Answer 2

The sequence converges to 1.

Step-by-step explanation:

Sequence 2/4, 3/5, 4/6, 5/7

This sequence can be summarized as:

`\sum_(n=0)^(\infty) (n+2)/(n+4)`

To test if it converges, we can calculate the limite of
`(n+2)/(n+4)` as n goes to infinite.

Limit:

`\lim_(n \rightarrow \infty) (n+2)/(n+4)`

Considering only the terms with the highest exponent in the numerator and the denominator:

`\lim_(n \rightarrow \infty) (n+2)/(n+4) = \lim_(n \rightarrow \infty) (n)/(n) = \lim_(n \rightarrow \infty) 1 = 1`

Thus, the sequences converges to 1.