Problem 28 - A Textbook of Differential Calculus

Question

Let \(a_{n}=\underbrace{2.99 \ldots 9,}_{n \text { times }} n \in N\) Statement I \(\left[\lim _{n \rightarrow \infty} a_{n}\right]=\lim _{n \rightarrow \infty}\left[a_{n}\right],[\cdot]\) denotes the greatest integer function. Statement II \(\lim _{n \rightarrow \infty} a_{n}=3\)

Step-by-Step Solution

Answer

Statement I is false, Statement II is true.

Step 1: Understanding the General Term

The sequence is defined as \( a_n \) where \( a_n = 2.99\ldots9 \) with the digit 9 repeating \( n \) times. Thus, each term \( a_n \) is a representation of a number approaching 3 as \( n \) increases, because 2.99... (with infinitely many 9's) is equal to 3.

Step 2: Analyzing Statement II

Statement II claims that \( \lim_{n \to \infty} a_n = 3 \). As we increase the number of 9's in \( a_n \), the value of \( a_n \) becomes closer and closer to 3. In fact, with infinitely many 9's following the 2, \( a_n \) effectively becomes 3. Thus, Statement II is true.

Step 3: Analyzing Statement I

Statement I involves the greatest integer function, denoted as \([x]\), which takes a real number \(x\) and returns the greatest integer less than or equal to \(x\). Since \( a_n \) is slightly less than 3 for any finite \( n \), \([a_n] = 2\). Thus, \( \lim_{n \to \infty} [a_n] = 2 \). However, \( \lim_{n \to \infty} a_n \) is 3, so \( \bigg[\lim_{n \to \infty} a_n\bigg] = [3] = 3 \). Therefore, Statement I is false, as the two sides do not equal each other.

Key Concepts

Greatest Integer Function

The Greatest Integer Function, often symbolized as \([x]\), is a mathematical method that finds the largest integer less than or equal to a given real number \(x\). It's sometimes referred to as the floor function. To understand this better, let's consider a few examples:

• For \(x = 2.7\), \([2.7] = 2\) because 2 is the greatest integer less than 2.7.
• For \(x = 4\), \([4] = 4\) since 4 is already an integer.
• For a negative number, like \(x = -1.2\), \([-1.2] = -2\) because -2 is the greatest integer smaller than -1.2.

The function effectively "rounds down" a number to its nearest whole number. In the context of our exercise, because each term \(a_n\) in the sequence is slightly less than 3 (think 2.999...), the greatest integer less than \(a_n\) is 2, regardless of how close \(a_n\) actually gets to 3.
Thus, the expression \([a_n]\) simplifies to 2 for any finite \(n\), even as \(n\) approaches infinity. This explains why the statement involving the greatest integer function is false in this particular situation.

Convergence of Sequences

In analysis, a sequence \(a_n\) "converges" if its terms tend to a specific value, known as the limit, as \(n\) becomes very large. A convergent sequence has a very important property: for any small number \(\epsilon\), no matter how tiny, there is a point beyond which all terms of the sequence lie within \(\epsilon\) of the limit.

For instance, consider a sequence that converges to 3. As the sequence progresses with larger and larger values of \(n\), it will eventually, and permanently, get as close to 3 as desired, within any \(\epsilon\). This idea of getting as close as needed is at the core of understanding convergence.

• In the given exercise, the sequence \(a_n\) where \(a_n = 2.999\ldots9\) (up to \(n\) times) illustrates this concept. The sequence gets increasingly close to 3 as \(n\) increases.
• This convergence is why \(\lim_{n \to \infty} a_n = 3\). Eventually, the sequence can be made to lie within any desired range of 3.

Hence, convergence offers a clear, precise way to say a sequence behaves as \(n\) goes to infinity.

Limits of Sequences

The concept of limits in sequences is foundational in calculus. The limit of a sequence is the value that its terms approach as the index \(n\) becomes indefinitely large. When we say "\(\lim_{n \to \infty} a_n = L\)", it means that the terms of the sequence \(a_n\) get arbitrarily close to \(L\) as \(n\) increases.

In mathematical terms, for any small positive number \(\epsilon\), there exists an integer \(N\) such that for all \(n > N\), the absolute difference \(|a_n - L| < \epsilon\).

• In the problem at hand, each \(a_n = 2.999\ldots9\) (with increasing nines reaching towards infinity) effectively targets the number 3 as its limit.
• As \(n\) increases without bound, the sequence terms come closer to 3, thus fulfilling the limit condition \(\lim_{n \to \infty} a_n = 3\).

The limit represents the long-term behavior of the sequence, allowing mathematicians to predict what happens as the sequence extends indefinitely.

Real Numbers

Real numbers are a fundamental concept in mathematics representing numbers on the number line, including both rational (such as fractions and integers) and irrational numbers (like \(\pi\) and \(\sqrt{2}\)). Every real number corresponds to a point on the number line, making them essential for understanding continuous variables and calculus.

• Rational numbers can be expressed as a ratio of two integers. For instance, 1/2, 3, and -4.7 are all real numbers.
• Irrational numbers cannot be expressed as simple fractions. They have non-repeating, non-terminating decimal expansions, such as the number represented by \(a_n = 2.99...9\).

In the context of the exercise, \(a_n\) represents a real number that gets closer to 3 without ever exactly becoming 3 for any finite \(n\).
The real number system is comprehensive and includes all the numbers encountered in high school math and beyond, making it robust for tackling a wide range of mathematical problems.