Use the graph of the function f of x equals x plus 1 over x to determine which of the following statements is true for the sequence a sub n equals the sequence n plus 1 over n . (15 points)
A) The sequence is monotonic and bounded.
B) The sequence is bounded but not convergent.
C) The sequence is unbounded but convergent.
D) The sequence is unbounded.
geez, it's a math question -- why all those words?
f(x) = (x+1)/x
f(x) = 1 + 1/x
the sequence is clearly monotonic and convergent (limit = 1)
Hmmm. Maybe you meant
f(x) = x + 1/x
That is clearly unbounded, since x→∞
Next time make it clear what you mean. Words do not do well expressing math, unless you are very careful.
Bottom line:
It's not my job to figure out what you mean. It's your job to say it so clearly you cannot be misunderstood.
To determine which statement is true for the sequence a subscript n equals n plus 1 over n, we can start by examining the graph of the function f(x) equals x plus 1 over x.
The graph of f(x) is a hyperbola that has a vertical asymptote at x equals 0 and a horizontal asymptote at y equals 1. This means that as x approaches infinity or negative infinity, the function approaches 1.
Now let's consider the sequence a subscript n equals n plus 1 over n. This sequence is a discrete representation of the function f(x) by substituting n for x. In other words, each term of the sequence is obtained by evaluating the function at a specific value of n.
Let's evaluate the sequence for a few values of n:
- For n equals 1, a(subscript 1) equals 1 plus 1 over 1, which is equal to 2.
- For n equals 2, a(subscript 2) equals 2 plus 1 over 2, which is equal to 2.5.
- For n equals 3, a(subscript 3) equals 3 plus 1 over 3, which is equal to 4/3, or approximately 1.33.
- For n equals 4, a(subscript 4) equals 4 plus 1 over 4, which is equal to 4.25.
As we can see, the values of the sequence increase as n increases. This tells us that the sequence is not monotonic, since it is not strictly increasing or strictly decreasing.
Next, let's consider whether the sequence is bounded or unbounded. The sequence is defined by adding 1 to n and then dividing by n. As n becomes larger and larger, the impact of adding 1 becomes less significant. Therefore, the sequence approaches 1 as n approaches infinity. This means that the sequence is bounded above by 1.
However, the sequence is not bounded below because there are no lower bounds. As n approaches infinity, the value of 1/n becomes arbitrarily close to 0, but it never reaches 0. Therefore, the sequence does not have a lower bound.
Based on our analysis, we can conclude that statement B) "The sequence is bounded but not convergent" is true for the sequence a subscript n equals n plus 1 over n. The sequence is bounded above by 1, but it does not converge to a single value. Instead, it approaches 1 as n becomes larger.