Question

Consider the sequence \(\sqrt{2}, \sqrt{2+\sqrt{2}}, \sqrt{2+\sqrt{2+\sqrt{2}}}, \ldots\) (a) Compute the first five terms of this sequence. (b) Write a recursion formula for \(a_{n}, n \geq 2\). (c) Find \(\lim _{n \rightarrow \infty} a_{n}\).

Short answer

The first five terms are \(a_1 = \sqrt{2}\), \(a_2 = \sqrt{2+\sqrt{2}}\), \(a_3 = \sqrt{2+\sqrt{2+\sqrt{2}}}\), \(a_4 = \sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2}}}}\), and \(a_5 = \sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2}}}}}\). The recursion formula is \(a_{n} = \sqrt{2+a_{n-1}}\) for \( n \geq 2\), and \( a_1 = \sqrt{2} \). The limit as \(n\) approaches infinity is \(2\).

Step-by-step solution

Compute First Five Terms

To find the first five terms of the sequence, you can simply substitute the given values of 'n' and calculate:\n\(a_1 = \sqrt{2}\)\n\(a_2 = \sqrt{2+\sqrt{2}}\)\n\(a_3 = \sqrt{2+\sqrt{2+\sqrt{2}}}\)\n\(a_4 = \sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2}}}}\)\n\(a_5 = \sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2}}}}}\)

Write a Recursive Formula

For a sequence with the given structure, the recursive formula can be expressed as: \n\(a_{n} = \sqrt{2+a_{n-1}}\) for \( n \geq 2\), and \( a_1 = \sqrt{2} \).

Find the Limit of \(a_{n}\) as \(n\) Approaches Infinity

Let \( L = \lim_{n \rightarrow \infty} a_{n} \). Write the recursive formula in terms of limit, \( L = \sqrt{2+L} \). Solving this for \( L \), yields the equation \( L^2 - L - 2 = 0 \). Solving the quadratic equation, one gets two solutions: \( L = -1, 2 \). Since elements of the sequence \( a_n \) are positive, the negative root is discarded. Hence, \( \lim_{n \rightarrow \infty} a_{n} = 2 \).

Key concepts

Convergence of Sequences

Understanding the concept of convergence of sequences is crucial in calculus. A sequence is said to converge if its terms approach a specific value, known as a limit, as the sequence progresses to infinity. This property is essential as it helps determine the behavior of sequences over a long stretch.

For example, consider the sequence given in the exercise: it comprises numbers constructed with nested square roots. We say this sequence converges if there is a real number that the terms get closer to as the index increases without bound. In the case of our sequence, as we compute more terms, it is observed that the numbers do appear to stabilize around a particular value. This is indicative of convergence. The importance of convergence lies in its ability to give insights into the stability and long-term behavior of sequences which is used in various fields including mathematics, physics, and engineering.

When learning about convergence, it's beneficial to also recognize when a sequence does not converge, also known as divergence. Sequences that oscillate or increase without limit are considered to diverge. Convergence in sequences is a stepping stone to understanding more complex concepts in calculus such as series and integrals.

Recursive Formula

A sequence can often be described by a recursive formula, which defines each term using one or more previous terms. It's like a domino effect; to find a specific piece, you need to start from the beginning and follow the chain.

In our example, the recursive formula for the sequence is defined as: \(a_{n} = \sqrt{2+a_{n-1}}\) for \( n \geq 2\), with the initial term \( a_1 = \sqrt{2} \). This is a simple yet powerful expression that allows us to generate any term in the sequence, as long as we know its predecessor.

Recursive formulas are not only essential for sequences but also for programming, where they are used to write functions that call themselves to solve a problem. They are also present in nature, as seen in patterns like fractals. Understanding recursive formulas can be a game-changer in mastering sequences and their applications.

Limit of a Sequence

The limit of a sequence is a fundamental concept in calculus that refers to the value the sequence's terms get closer to as the index goes to infinity. If such a limit exists, we can say that the sequence converges to that particular value.

For instance, when we say that the limit of the sequence \(a_n\) as \(n\) approaches infinity is \(L\), we write \(\lim _{n \rightarrow \infty} a_{n} = L\). In the step-by-step solution, by recognizing that the recursive sequence approaches a limit \(L\), we derive \(L = \sqrt{2+L}\) and solve to find \(L = 2\), since it's the only positive solution that makes sense in the context.

The concept of a limit is not only restricted to sequences but extends to functions as well. It has implications in defining continuity, derivatives, and integrals, which are central concepts in calculus. The limit concept aids in mathematical rigor by allowing mathematicians to talk about the infinitely small or large in a precise way.