Proving A_n < 2 And Finding The Limit Of The Sequence A_n = √(2 + A_{n-1})

Introduction

In this article, we delve into the fascinating world of sequences and explore the properties of a specific sequence defined recursively. Our focus is on the sequence where the initial term, a_0, is equal to the square root of 2 (√2), and each subsequent term, a_{n+1}, is obtained by taking the square root of the sum of 2 and the preceding term (√(2 + a_n)). We aim to rigorously prove that every term in this sequence remains strictly less than 2 for all finite values of n. Furthermore, we will embark on a journey to determine the limit of this sequence as n approaches infinity, unraveling the value the sequence converges to.

This exploration will not only enhance our understanding of sequences and their behavior but also provide insights into the powerful techniques used in mathematical analysis, such as mathematical induction and limit calculations. By meticulously examining the sequence's properties, we will gain a deeper appreciation for the elegance and precision inherent in mathematical reasoning.

Proof of a_n < 2 for all finite n

To demonstrate that a_n < 2 for all finite n, we will employ the powerful technique of mathematical induction. Mathematical induction is a fundamental method for proving statements that hold for all natural numbers. It involves two key steps: the base case and the inductive step.

Base Case (n = 0):

We begin by establishing the base case, which is the foundation of our inductive argument. When n = 0, we have a_0 = √2. Since the square root of 2 is approximately 1.414, which is indeed less than 2, the base case holds true. This provides us with the initial foothold for our inductive proof.

Inductive Hypothesis:

Next, we assume that the statement holds for some arbitrary natural number k. This assumption is known as the inductive hypothesis. In our context, we assume that a_k < 2 for some k ≥ 0. This assumption is crucial as it forms the basis for our inductive step.

Inductive Step:

The heart of mathematical induction lies in the inductive step. Here, we aim to show that if the statement holds for k, it must also hold for k + 1. In other words, we want to prove that if a_k < 2, then a_{k+1} < 2.

Recall that a_{k+1} = √(2 + a_k). Since we have assumed that a_k < 2, we can add 2 to both sides of the inequality, resulting in 2 + a_k < 4. Now, taking the square root of both sides (and noting that the square root function is monotonically increasing for non-negative numbers), we get √(2 + a_k) < √4, which simplifies to a_{k+1} < 2.

This completes the inductive step. We have successfully shown that if the statement holds for k, it also holds for k + 1.

Conclusion by Mathematical Induction:

By the principle of mathematical induction, since the base case holds and the inductive step is proven, we can confidently conclude that a_n < 2 for all finite natural numbers n. This elegant proof demonstrates the power of mathematical induction in establishing statements that hold across an infinite range of values.

Finding the Limit of a_n as n approaches infinity

Now that we have established that the sequence a_n is bounded above by 2, we turn our attention to determining its limit as n approaches infinity. To achieve this, we will first demonstrate that the sequence is monotonically increasing.

Proof of Monotonic Increase:

To show that the sequence is monotonically increasing, we need to prove that a_{n+1} > a_n for all n ≥ 0. Let's consider the difference between a_{n+1} and a_n:

a_{n+1} - a_n = √(2 + a_n) - a_n

To analyze the sign of this difference, we can manipulate the expression by multiplying and dividing by the conjugate:

[√(2 + a_n) - a_n] * [√(2 + a_n) + a_n] / [√(2 + a_n) + a_n] = (2 + a_n - a_n²) / [√(2 + a_n) + a_n]

Now, let's focus on the numerator: 2 + a_n - a_n². This is a quadratic expression in a_n. We can rewrite it as -(a_n² - a_n - 2), which can be further factored as -(a_n - 2)(a_n + 1).

Since we have already proven that a_n < 2 for all n, the factor (a_n - 2) is always negative. The factor (a_n + 1) is always positive because a_n is the square root of a number and therefore non-negative. Thus, the numerator -(a_n - 2)(a_n + 1) is always positive.

The denominator, √(2 + a_n) + a_n, is also always positive since it's the sum of a square root and a non-negative number.

Therefore, the entire expression (2 + a_n - a_n²) / [√(2 + a_n) + a_n] is positive, which means a_{n+1} - a_n > 0, and consequently, a_{n+1} > a_n. This confirms that the sequence is monotonically increasing.

Applying the Monotone Convergence Theorem:

We have now established two crucial properties of the sequence a_n: it is bounded above (by 2) and monotonically increasing. These are the precise conditions required for the Monotone Convergence Theorem to apply.

The Monotone Convergence Theorem states that if a sequence is both bounded and monotonic (either increasing or decreasing), then it must converge to a limit. In our case, since a_n is bounded above and monotonically increasing, we can confidently conclude that it converges to a limit, which we will denote as L.

Calculating the Limit L:

To find the value of L, we take the limit of both sides of the recursive definition a_{n+1} = √(2 + a_n) as n approaches infinity:

lim (n→∞) a_{n+1} = lim (n→∞) √(2 + a_n)

Since the limit of a_{n+1} as n approaches infinity is the same as the limit of a_n, we can replace both with L:

L = √(2 + L)

To solve for L, we square both sides of the equation:

L² = 2 + L

Rearranging the terms, we get a quadratic equation:

L² - L - 2 = 0

This quadratic equation can be factored as:

(L - 2)(L + 1) = 0

This gives us two possible solutions for L: L = 2 and L = -1. However, since all the terms in the sequence a_n are positive (as they are square roots), the limit L must also be non-negative. Therefore, we discard the solution L = -1 and conclude that the limit of the sequence is L = 2.

Conclusion

In summary, we have rigorously proven that for the sequence defined by a_0 = √2 and a_{n+1} = √(2 + a_n), all terms are strictly less than 2 for all finite n. Furthermore, we have demonstrated that the sequence is monotonically increasing and converges to a limit of 2 as n approaches infinity. This exploration highlights the power of mathematical induction and the Monotone Convergence Theorem in analyzing the behavior of sequences. By combining these tools, we have gained a deep understanding of the properties and limiting behavior of this particular sequence.

This analysis not only provides a concrete example of sequence convergence but also reinforces the importance of mathematical rigor in establishing the validity of mathematical statements. The techniques employed here can be extended to analyze a wide range of sequences and functions, making this a valuable exercise in mathematical understanding.