Question

Consider the following sequences defined by a recurrence relation. Use a calculator, analytical methods, and/or graphing to make a conjecture about the limit of the sequence or state that the sequence diverges. $$a_{n+1}=\sqrt{2+a_{n}} ; a_{0}=1$$

Short answer

Based on the given recurrence relation $$a_{n+1}=\sqrt{2+a_{n}}$$ with $$a_{0}=1$$, and by generating several terms, the sequence appears to be increasing and approaching a limit, denoted as \(L\). By solving the quadratic equation for the limit, we obtain two possible solutions \(L_1=2\) and \(L_2=-1\). Since the sequence is positive and increasing, we can conclude that the correct limit is $$L=L_1=2$$.

Step-by-step solution

Understand the Recurrence Relation

The given recurrence relation is $$a_{n+1}=\sqrt{2+a_{n}}$$, with $$a_{0}=1$$. The sequence is generated by plugging the previous term into the given formula. For instance, using $$a_{0}=1$$ to find $$a_{1}$$, $$a_{1} = \sqrt{2 + a_{0}} = \sqrt{2 + 1} = \sqrt{3}$$. Similarly, to find $$a_{2}$$, use $$a_{1}$$, $$a_{2} = \sqrt{2 + a_{1}} = \sqrt{2 + \sqrt{3}}$$.

Generate Several Terms and Observe the Behavior

Calculate a few more terms of the sequence and observe the behavior. $$a_{3} = \sqrt{2 + a_{2}} = \sqrt{2 + \sqrt{2 + \sqrt{3}}}$$ $$a_{4} = \sqrt{2 + a_{3}} = \sqrt{2 + \sqrt{2 + \sqrt{2 + \sqrt{3}}}}$$ It seems difficult to observe a pattern from the expressions. We can use a calculator to approximate the values: $$a_{1} ≈ 1.73205$$ $$a_{2} ≈ 1.93185$$ $$a_{3} ≈ 1.98289$$ The sequence appears to be increasing and approaching a certain value. Let's denote this limit as \(L\).

Define the Limit

Assume that the sequence converges to its limit \(L\). We then have the following relationship: $$L = \sqrt{2 + L}$$ Our goal is to determine the value of \(L\).

Solve for the Limit

To find the limit \(L\), solve the equation: $$L = \sqrt{2 + L}$$ Square both sides: $$L^2 = 2 + L$$ Rearranging, we get a quadratic equation: $$L^2 - L - 2 = 0$$ Now, apply the quadratic formula, where \(a=1\), \(b=-1\), and \(c=-2\): $$L=\frac{-b\pm\sqrt{b^2-4ac}}{2a}=\frac{1\pm\sqrt{1^2-4\cdot1\cdot(-2)}}{2\cdot 1}=\frac{1\pm\sqrt{9}}{2}$$ This yields two possible solutions: $$L_1=\frac{1+3}{2}=2$$ $$L_2=\frac{1-3}{2}=-1$$

Choose the Correct Limit

We can eliminate one of the solutions by observing the sequence's behavior. Since our sequence values are positive and increasing, we can conclude that the correct limit is: $$L=L_1=2$$ Therefore, the limit of the given recurrence relation is 2.

Key concepts

Convergence of Sequences

In the mathematical field of calculus, understanding the convergence of sequences is crucial. A sequence is considered to converge if its terms approach a specific value, called the limit, as the number of terms increases indefinitely. To visualize this, think of a sequence as a line of stepping stones across a river. If you can step from one stone to another and eventually reach one final stone without having to make infinite steps, then that final stone represents the limit and the sequence converges.

To determine whether a given sequence converges, we can observe its behavior as we calculate more and more terms. If there's a noticeable pattern where the terms are getting closer and closer to a particular number, it's likely that the sequence converges. In the context of our example, where the sequence is defined recursively by the relation ( a_{n+1}=d{d{2+(n}))) with a starting value of (n_{0}=1)), we observe that the numerical values of the terms seem to increase and approach a number. This suggests that the sequence may indeed have a limit to which it converges.

Solving Recurrence Relations

Solving recurrence relations means finding a function or formula that can describe the terms of a sequence without the need to refer back to previous terms. Recurrence relations can often be complex and intimidating, but they are essentially just puzzles waiting to be solved. A common method of tackling these puzzles is by guessing and then proving a formula, or by transforming the recursive formula into a non-recursive one.

In our case, we have a recurrence relation (n_{n+1}=d{d{2+(n}}). Instead of calculating each term using the previous one, we look at the pattern and infer a regular behavior as done in the original solution. By assuming the sequence has a limit (L)), which it approaches, and setting up the equation (L = d{d{2+L})), we effectively transform the recurrence relation into an algebraic equation. This allows us to solve for (L)) analytically, providing us with a deeper understanding of the sequence and how it behaves in the limit.

Limits of Sequences

The limit of a sequence is a fundamental concept in calculus that represents the value a sequence approaches as the number of terms goes to infinity. It's not about the sequence ever reaching that value but rather getting closer and closer to it as it progresses. To determine the limit, if it exists, we look for a value that each term of the sequence can get arbitrarily close to as the sequence goes on indefinitely.

In our example, we have already anticipated the limit by examining the behavior of the initial terms and setting up the equation (L = d{d{2+L})), which we derived from the recurrence relation. After solving the quadratic equation obtained from the limit equation, we're left with two possible limits, (L_1=2)) and (L_2=-1)). To arrive at the correct limit, we applied the sequence's behavior: it's positive and increasing. Hence, the correct limit that fits the context of our sequence is (L=2)). This approach links the abstract concept of a limit to the concrete values and behavior of the sequence we've been analyzing.