Question

Prove that if \(\left\\{a_{n}\right\\} \ll\left\\{b_{n}\right\\}\) (as used in Theorem 8.6 ), then \(\left\\{c a_{n}\right\\} \ll\left\\{d b_{n}\right\\},\) where \(c\) and \(d\) are positive real numbers.

Short answer

Question: Prove that if the sequence {𝑎𝑛} is negligible in comparison to the sequence {𝑏𝑛}, then the sequence {𝑐𝑎𝑛} is negligible in comparison to the sequence {𝑑𝑏𝑛}, where 𝑐 and 𝑑 are positive real numbers. Answer: This can be proven by showing that if \(\lim_{n \to \infty}\frac{a_n}{b_n} = 0\), then \(\lim_{n \to \infty}\frac{ca_n}{db_n} = 0\). Since we found that \(\lim_{n \to \infty}\frac{ca_n}{db_n} = 0\), it means that the sequence {𝑐𝑎𝑛} is negligible in comparison to the sequence {𝑑𝑏𝑛}, which proves the given statement.

Step-by-step solution

Understand the definition of the negligible sequence

A sequence \(\{a_n\}\) is negligible in comparison to the sequence \(\{b_n\}\) (denoted as \(\{a_n\} \ll \{b_n\}\)) if and only if \(\lim_{n \to \infty}\frac{a_n}{b_n} = 0\). In other words, the ratio \(\frac{a_n}{b_n}\) approaches zero as \(n\) goes to infinity.

Recall the given condition

We are given that \(\{a_n\} \ll \{b_n\}\). According to the definition of the negligible sequence, we can write this as: \(\lim_{n \to \infty}\frac{a_n}{b_n} = 0\).

Find the limit of the ratio of the modified sequences

Now, we have to find the limit of the ratio \(\frac{ca_n}{db_n}\) as \(n\) goes to infinity. This can be done as follows: $$ \lim_{n \to \infty}\frac{ca_n}{db_n} = \frac{c}{d} \lim_{n \to \infty}\frac{a_n}{b_n} $$

Apply the given condition

Since we know that \(\lim_{n \to \infty}\frac{a_n}{b_n} = 0\), the limit of the modified sequences will be: $$ \lim_{n \to \infty}\frac{ca_n}{db_n} = \frac{c}{d} (0) = 0 $$

Prove the statement

Since we found that \(\lim_{n \to \infty}\frac{ca_n}{db_n} = 0\), it means that the sequence \(\{ca_n\}\) is negligible in comparison to the sequence \(\{db_n\}\). Therefore, we can conclude that \(\{ca_n\} \ll \{db_n\}\), which proves the given statement.

Key concepts

Limits of Sequences

When studying calculus, one of the foundational concepts you will encounter is the limit of a sequence. Let's dive into what this means, using simple terms. Imagine a list of numbers, which we call a sequence. Each number in this list is called a term of the sequence, and you can think of these numbers as a line of people waiting to compete in a race.

Now, if you were to look at this line from a distance, you might not be able to distinguish each individual runner, especially as they get further away from you. Similarly, when mathematicians talk about the limit of a sequence, they are considering what value the terms of the sequence are approaching as they go on indefinitely, just like the line of racers stretching out into the distance. In technical terms, we say the limit of the sequence \( \{a_n\} \) as \( n \) approaches infinity, written \( \lim_{{n \to \infty}} a_n \), is a certain value \( L \) if the terms \( a_n \) get closer and closer to \( L \) as \( n \) becomes larger and larger.

When someone says a sequence is negligible, it means that as the sequence progresses, its terms get closer and closer to zero. Mathematically, it's like saying, 'No matter how hard you squint, that line of people eventually looks like it disappears.' In our exercise example, we see this concept in action as the ratio of the sequences \( \frac{{a_n}}{{b_n}} \) tends toward zero, indicating that \( \{a_n\} \) is indeed negligible in comparison to \( \{b_n\} \) as they race towards infinity.

Sequence Comparison

Sequence comparison is akin to putting two runners side by side and seeing who is faster—it's all about comparing the behavior of one sequence relative to another. In the realm of calculus, we use sequence comparison to understand the relationship between two sequences as they head towards infinity.

Let's say we have two sequences, \( \{a_n\} \) and \( \{b_n\} \)—imagine them as two different lines of racers. When we write \( \{a_n\} \ll \{b_n\} \) (as seen in our exercise), it's like declaring that the runners in line \( \{a_n\} \) are so much slower that they are practically standing still compared to line \( \{b_n\} \)—their relative pace is approaching zero.

Mathematically, to compare sequences, we often look at the limit of their ratio. If the limit of \( \frac{{a_n}}{{b_n}} \) is zero as \( n \) becomes infinitely large, it tells us that \( \{a_n\} \) is becoming negligible compared to \( \{b_n\} \) at a fast rate. Through this exercise, we're extending that comparison to altered sequences—with constants \( c \) and \( d \)—and finding that the relationship of negligibility is maintained.

Infinity in Calculus

In calculus, infinity is not a number but rather a concept—it's the idea of something with no end, like a racetrack that loops around the earth and keeps going forever. When we talk about sequences approaching infinity, we're discussing the behavior of these sequences as the terms grow without bounds.

Understanding Infinity Through Limits

In our example, when evaluating the limit \( \lim_{{n \to \infty}} \frac{{ca_n}}{{db_n}} \), we're watching our racers approach an infinite racetrack. They can keep running forever, and even if we multiply their speed by any positive number (as seen with the constants \( c \) and \( d \) in the exercise), their relative speed can still tell us a lot about their behavior. If one sequence grows much more quickly than the other, such as \( \{b_n\} \) outpacing \( \{a_n\} \) by a significant margin, this swiftly becomes evident on our infinite racetrack.

Infinity often seems daunting because it is difficult to comprehend something without limits, much like the horizon: you can keep heading towards it, but the exact spot where the sky meets the earth is never actually reached. In calculus, we use infinity to describe the idea of going 'as far as you can imagine and then some', always moving forward, without ever stopping.