Question

Is the sequence 2/3, 3/4, 4/5, 5/6... (n+1)/(n+2) divergent

Answer 1

To see if a sequence converges, you have to show that it is monotonic and bounded. If the sequence fails to meet either condition, it diverges.

Boundedness is easy to see. As \(n\) increases indefinitely, the expression \(\dfrac{n+1}{n+2}\) gets arbitrarily closer to 1 (without attaining 1, since the numerator is always smaller than the denominator). You might be expected to prove this formally.

Showing monotonicity involves showing that a given term is smaller or larger than its successor. In other words, show that \(a_n\le a_{n+1}\) for all \(n\). This is also fairly easy to see if you consider the decimal values of each fraction (.667, .75, .8, ...).

Answer 2

okay thanks so it is convergent then.

Answer 3

Yes

Answer 4

thanks

Answer 5

yw