Sequence Analysis: Inf, Sup, Limit, And Limit Superior

Let's dive into analyzing sequences, guys! We're going to tackle finding the infimum (inf), supremum (sup), limit ($\lim_{n\to\infty} x_n$), and limit superior ($\limsup_{n\to\infty} x_n$) for a few given sequences. This might sound intimidating, but we'll break it down step by step. Think of it like this: we're trying to understand the overall behavior of these sequences as they go on forever. Where do they tend to settle, and what are their boundaries?

Understanding the Concepts

Before we jump into the calculations, let’s make sure we're all on the same page about what these terms mean. It’s super important to have a solid grasp of these concepts before tackling the problems.

• Infimum (inf $x_n$): The infimum is the greatest lower bound of a sequence. Think of it as the largest number that's less than or equal to every term in the sequence. It's like the floor of the sequence. The infimum helps us define a sequence’s lower boundary. Even if a sequence never actually reaches a specific value, the infimum tells us the greatest value it never goes below. In simpler terms, if you’re looking at a bunch of numbers, the infimum is the biggest number that's still smaller than or equal to all the other numbers.
• Supremum (sup $x_n$): On the flip side, the supremum is the least upper bound. It's the smallest number that's greater than or equal to every term in the sequence. It's like the ceiling of the sequence. The supremum is equally important as it shows the sequence’s upper boundary. Just like the infimum, the supremum might not be a value the sequence ever touches, but it's the smallest value that the sequence never exceeds. Imagine the supremum as the smallest number that’s still bigger than or equal to all the numbers in a sequence.
• Limit ($\lim_{n\to\infty} x_n$): The limit is the value that the sequence approaches as n (the term number) gets larger and larger – approaching infinity. If the sequence has a limit, the terms get arbitrarily close to this value. Not all sequences have limits; some might oscillate or grow without bound. The limit describes where a sequence 'settles down'. If a sequence has a limit, it means that as you go further and further along the sequence, the terms get closer and closer to a particular value. It's like the sequence is aiming for a target.
• Limit Superior ($\limsup_{n\to\infty} x_n$): The limit superior is a bit trickier. It's the largest limit of any convergent subsequence. In simpler terms, if you pick out parts of the original sequence that do converge, the limit superior is the largest value those parts converge to. If the sequence converges, the limit superior is the same as the limit. The limit superior, often abbreviated as lim sup, is a crucial concept for sequences that don't have a single, clear limit. It's like finding the highest 'target' the sequence aims for, even if it doesn't always hit it. The limit superior is found by considering all the subsequences (parts of the sequence) that do converge and then picking out the largest limit among them. If a sequence converges, its limit superior is simply its limit.

Problem 99: $x_n = n^2 - 9n - 100$

Let's get our hands dirty with the first sequence: $x_n = n^2 - 9n - 100$. Our goal here is to find the infimum, supremum, limit, and limit superior.

Analyzing the Sequence

First, let's think about what this sequence looks like. It's a quadratic function with a positive leading coefficient ($n^2$), so it forms a parabola opening upwards. This means that as n gets very large, the terms will also get very large in the positive direction. This gives us a clue about the limit and supremum. Understanding the underlying function is crucial. In this case, recognizing the quadratic nature of $x_n = n^2 - 9n - 100$ is the key. Since the coefficient of $n^2$ is positive, the parabola opens upwards. This immediately tells us that as n gets larger, $x_n$ will also get larger without bound, which helps us determine the limit and supremum.

Finding the Infimum

To find the infimum, we need to figure out the lowest point of the parabola. We can do this by finding the vertex of the parabola. The x-coordinate (in our case, the n-coordinate) of the vertex is given by $-b/2a$, where a and b are the coefficients of the quadratic equation. Here, $a = 1$ and $b = -9$, so the vertex occurs at $n = -(-9) / (2 * 1) = 4.5$. Since n must be an integer, we need to check the values of $x_n$ for the integers around 4.5, which are $n = 4$ and $n = 5$.

Let's calculate $x_4$ and $x_5$:

• $x_4 = 4^2 - 9(4) - 100 = 16 - 36 - 100 = -120$
• $x_5 = 5^2 - 9(5) - 100 = 25 - 45 - 100 = -120$

It seems like -120 might be the infimum, but we should check a few more values to be sure. Let's check $x_9$ and $x_{10}$:

• $x_9 = 9^2 - 9(9) - 100 = 81 - 81 - 100 = -100$
• $x_{10} = 10^2 - 9(10) - 100 = 100 - 90 - 100 = -90$

Now, let’s calculate the values of $x_n$ for $n = 4$ and $n = 5$ to pinpoint the minimum value. These integer values around the vertex will likely give us the infimum. We find that $x_4 = -120$ and $x_5 = -120$. So, the infimum of the sequence is -120.

So, the infimum is -120. The infimum of a sequence indicates its greatest lower bound. In this case, by finding the vertex of the parabola defined by $x_n$, we identify the potential minimum value. Calculating $x_4$ and $x_5$ confirms that -120 is indeed the infimum, as it is the smallest value the sequence attains.

Finding the Supremum

Since the parabola opens upwards and the terms keep increasing as n increases, there is no upper bound. Therefore, the supremum is positive infinity ($\infty$). The supremum for this sequence is positive infinity. Given the parabola opens upwards, the sequence grows without bound as $n$ increases. This signifies there is no finite upper bound, leading us to conclude the supremum is positive infinity.

Finding the Limit

As n approaches infinity, $n^2$ dominates the other terms in the expression, so the sequence goes to infinity. Thus, the limit is positive infinity ($\infty$). Determining the limit involves observing the sequence's behavior as $n$ approaches infinity. In the case of $x_n = n^2 - 9n - 100$, the $n^2$ term significantly outweighs the others as $n$ grows, causing the sequence to tend towards positive infinity.

Finding the Limit Superior

Since the sequence diverges to infinity, the limit superior is also positive infinity ($\infty$). The limit superior coincides with the limit when a sequence diverges to infinity. This reinforces that as $n$ increases, the sequence does not approach any finite value but continues to grow indefinitely.

Summary for Problem 99

• Inf $x_n$ = -120
• Sup $x_n$ = $\infty$
• $\lim_{n\to\infty} x_n$ = $\infty$
• $\limsup_{n\to\infty} x_n$ = $\infty$

Problem 100: $x_n = n + \frac{100}{n}$

Now, let’s tackle the sequence $x_n = n + \frac{100}{n}$. This one is a little different, and we'll need to think about how the two terms interact as n changes.

Analyzing the Sequence

We have two terms here: n, which increases as n increases, and $\frac{100}{n}$, which decreases as n increases. We need to find out how these two terms balance each other out. As n gets very large, $\frac{100}{n}$ approaches zero, and the sequence will behave more like n. This gives us a hint about the limit and supremum. Analyzing this sequence involves understanding the interplay between the two terms: $n$ and $\frac{100}{n}$. As $n$ grows, the term $\frac{100}{n}$ diminishes, while $n$ increases. This dynamic is vital for determining the sequence's long-term behavior.

Finding the Infimum

To find the infimum, we can consider the function $f(x) = x + \frac{100}{x}$ for real numbers x. We can find the minimum by taking the derivative and setting it equal to zero:

$f'(x) = 1 - \frac{100}{x^2}$

Setting $f'(x) = 0$, we get:

$1 - \frac{100}{x^2} = 0$

$\frac{100}{x^2} = 1$

$x^2 = 100$

$x = \pm 10$

Since n is a positive integer, we only consider $x = 10$. To pinpoint the infimum, we can treat the sequence as a continuous function and use calculus. Taking the derivative of f(x) = x + rac{100}{x} and setting it to zero helps us find the critical points. We find a critical point at $x = 10$, which is crucial for determining the sequence's minimum value.

Now we need to check the value of $x_n$ around $n = 10$. Let's calculate $x_{10}$, $x_9$, and $x_{11}$:

• $x_{10} = 10 + \frac{100}{10} = 10 + 10 = 20$
• $x_9 = 9 + \frac{100}{9} \approx 9 + 11.11 = 20.11$
• $x_{11} = 11 + \frac{100}{11} \approx 11 + 9.09 = 20.09$

It looks like $x_{10} = 20$ is the minimum value. Checking the values of $x_n$ around $n = 10$, specifically $x_9$, $x_{10}$, and $x_{11}$, we find that $x_{10} = 20$ is the smallest value. This confirms that 20 is the infimum of the sequence.

So, the infimum is 20.

Finding the Supremum

As n gets very large, the term n dominates, and $\frac{100}{n}$ approaches zero. Therefore, the sequence goes to infinity, and the supremum is positive infinity ($\infty$). The supremum of the sequence is positive infinity. As $n$ increases, the term $n$ becomes dominant, causing the sequence to grow without bound. This indicates that there is no finite upper limit, hence the supremum is positive infinity.

Finding the Limit

Similarly, as n approaches infinity, the limit is also positive infinity ($\infty$). The limit as $n$ approaches infinity is also positive infinity. The behavior of the sequence as $n$ becomes very large is governed by the term $n$, which tends towards infinity. This leads to the sequence having a limit of positive infinity.

Finding the Limit Superior

Since the sequence diverges to infinity, the limit superior is also positive infinity ($\infty$). The limit superior, similar to the limit, is positive infinity. The sequence's divergence to infinity implies that the limit superior also approaches infinity, reinforcing the absence of a finite upper bound.

Summary for Problem 100

• Inf $x_n$ = 20
• Sup $x_n$ = $\infty$
• $\lim_{n\to\infty} x_n$ = $\infty$
• $\limsup_{n\to\infty} x_n$ = $\infty$

Problem 101: $x_n = 1 - \frac{1}{n^2}$

Finally, let's analyze the sequence $x_n = 1 - \frac{1}{n^2}$. This sequence involves a fraction that decreases as n increases, so it should have a finite limit.

Analyzing the Sequence

As n gets larger, $\frac{1}{n^2}$ gets smaller and approaches zero. Therefore, $x_n$ approaches 1. This gives us a good idea of what the limit will be. The key to analyzing this sequence lies in understanding how $\frac{1}{n^2}$ behaves as $n$ grows. This term diminishes, allowing $x_n$ to approach 1, which is a crucial insight for determining the sequence's limit and supremum.

Finding the Infimum

The smallest value of $x_n$ occurs when n is the smallest, which is $n = 1$. So, $x_1 = 1 - \frac{1}{1^2} = 1 - 1 = 0$. To find the infimum, consider the smallest value of $n$, which is 1. Substituting $n = 1$ into the sequence, we get $x_1 = 0$, indicating that 0 is the smallest value the sequence attains and thus the infimum.

Therefore, the infimum is 0.

Finding the Supremum

As n approaches infinity, $\frac{1}{n^2}$ approaches 0, so $x_n$ approaches 1. Since $x_n$ is always less than 1, the supremum is 1. The supremum is determined by observing the sequence's upper bound. As $n$ approaches infinity, $x_n$ approaches 1, and since $x_n$ is always less than 1, the supremum is 1.

Finding the Limit

The limit as n approaches infinity is 1, as we discussed earlier. The limit of the sequence as $n$ approaches infinity is 1. As $n$ becomes very large, the term $\frac{1}{n^2}$ becomes negligible, causing $x_n$ to converge to 1.

Finding the Limit Superior

Since the sequence converges to 1, the limit superior is also 1. The limit superior, in this case, is also 1. Since the sequence converges to 1, the limit superior and the limit coincide.

Summary for Problem 101

• Inf $x_n$ = 0
• Sup $x_n$ = 1
• $\lim_{n\to\infty} x_n$ = 1
• $\limsup_{n\to\infty} x_n$ = 1

Conclusion

We've successfully analyzed three sequences, finding their infimum, supremum, limit, and limit superior. Remember, guys, the key is to understand the behavior of the sequence as n gets large and to think about the definitions of these concepts. Keep practicing, and you'll become a sequence analysis pro in no time!