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.
So I found that the sequence is descending changing the n's for x's and using the derivative. But now I'm not sure on how to proceed to find either of the bounds.
so I realized it's not using the derivative that we find if a sequenec is ascending/descending, but rather comparing the series. Nonetheless, the sequence is still descending.
To determine if a sequence is ascending or descending, you can examine the behavior of its terms. If each term is greater than the previous term, the sequence is ascending. If each term is smaller than the previous term, the sequence is descending.
To find the lower bound (b) or upper bound (B) of a sequence, you need to determine the limiting behavior of the sequence as n approaches positive infinity.
Let's start by analyzing the sequence {((n + 1)^2) / (n^2 + 1)}. To determine if it is ascending or descending, we need to compare subsequent terms. Let's examine the ratio of the (n+1)th term to the nth term:
((n+1)^2) / (n^2 + 1) = (n^2 + 2n + 1) / (n^2 + 1)
Simplifying further:
(n^2 + 2n + 1) / (n^2 + 1) = 1 + (2n / (n^2 + 1))
Now, let's analyze the behavior of this expression as n approaches positive infinity. As n gets larger and larger, the term (2n / (n^2 + 1)) becomes very close to zero. Consequently, the entire expression approaches 1.
Therefore, since the expression approaches 1, the terms of the sequence are decreasing as n increases. This means that the sequence is descending.
Next, let's determine if there is a lower bound (b) or upper bound (B) of the sequence. In this case, as n approaches positive infinity, we can see that the terms of the sequence approach but never reach 1. Therefore, there is no lower bound (b) or upper bound (B) for the sequence.
In summary, the given sequence {((n + 1)^2) / (n^2 + 1)} is descending and does not have a lower bound or upper bound.