Determine if the following sequence:
{ (n + 1)^2 / (n^2 + 1) },
is ascending, descending and find the lower bound b OR the upper bound B.
(n + 1)^2 / (n^2 + 1)
>
((n+1) + 1)^2 / ((n+1)^2 + 1)
so the sequence is descending,
it's the bounds I don't know how to find.
(n+1)^2/(n^2+1) = 1 + 2n/(n^2+1)
the sequence obviously increases, and the limit is 1
but if I put n = 1 in both I get =2 for the first and =1.8 for n+1. Or is that not how I go about doing it?
Oops. The sequence decreases, since 2n < n^2+1
The sequence goes 2, 1.8, 1.6, 1.47, 1.39, 1.32 ... converging to 1
so the upper bound is 1, since it converges, right?
let's see. every term is greater than 1. Why would you say 1 is the upper bound?
1 is the lower bound, no?
Because I thought that if it decreases it'll eventually get lower than 1. What is the definition of lower and upper bound, I don't get that part.
Oh, 2 would be the upper bound? Because it can't go higher than 2?
or rather it starts off at 2
note that each term is 1 + a fraction.
as the fraction decreases, the terms approach 1, but are always slightly greater than 1.
Thus, the minimum value achievable is 1, which makes 1 a lower bound. In fact, it is the greatest lower bound: there is no number greater than 1 which is less than all terms of the sequence.