Given the sequence, write an equation (Calculus 2)

Question

Given the sequence 3/5, 2/3, 5/7, 3/4, 7/9

a) write the equation for the sequence

b) does the sequence diverge or converge?

Answer 1

The denominator goes 5,3,7,4,9

which are consecutive odds beginning with 5,

followed by consecutive integers beginning with 3.

So the sequence for the denominator is: 5,3,7,4,9,5,11,6,13,7,15,8, 17, 9, 19, 10, 21, 11, 23, 12

The numerator differs from the denominator by 2, then 1, then 2, then 1, etc.

So here is the sequence:

3/5, 2/3, 5/7, 3/4, 7/9, 4/5, 9/11, 5/6, 11/13, 6/7, 13/15, 7/8, 15/17, 8/9, 17/19, 9/10, 19/21, 10/11, 21/23, 11/12

The odd index terms: n/(n+2) which by L'Hopital's rule converges to 1

The even index terms: n/(n+1) also converges to 1

So it converges to 1

Answer 2

At a glance it may look as though the terms of even index follow a different pattern than those of odd index. But since 2/3=4/6 and 3/4=6/8 and so on, the sequence can also be written 3/5, 4/6, 5/7, 6/8, 7/9, ...

So the sequence is simply a_n = n/(n+2) which holds for all n ≥ 3. Letting n go to infinity, a_n goes to 1.

Answer 3

Two sequences are interleaved:

3/5, 5/7, 7/9, ... (2n+1) / (2n + 3)

2/3, 3/4. 4/5. ... (2n) / (2n + 1)

Both converge to 0.