Problem 21 - Advanced Calculus

Question

Let \(a_{1}=0, a_{2}=1\), and \(a_{n+2}=\frac{n a_{n+1}+a_{n}}{n+1}\) (a) Calculate the value of \(a_{6}\) and \(a_{7}\). (b) Prove that \(\left\\{a_{n}\right\\}\) con verges. * \((c)\) Show that \(\lim _{n \rightarrow x} a_{n}=1-e^{-1}\).

Step-by-Step Solution

Answer

The value of \(a_6\) is \(1/5\) and \(a_7\) is \(1/6\). The sequence \(\left\{a_{n}\right\}\) is proven to converge. And the limit of \(a_{n}\) as \(n\) approaches infinity is \(1 - e^{-1}\).

Step 1: Calculate a6 and a7

Firstly, use the given recursive formula to find the values of \(a_6\) and \(a_7\). Start from the given initial conditions \(a_1\) and \(a_2\) and iterate the sequence using the recursive definition.\n\nSo, let's calculate \n\n\(a_{n+2}=\frac{n a_{n+1}+a_{n}}{n+1}\).\n\nThen, \(a_3=\frac{2*a_2+a_1}{2}=1/2\),\n\n\(a_4=\frac{3*a_3+a_2}{3} = 1/3\),\n\n\(a_5=\frac{4*a_4+a_3}{4}= 1/4\),\n\n\(a_6=\frac{5*a_5+a_4}{5}=1/5\),\n\n\(a_7= \frac{6*a_6+a_5}{6}=1/6\).

Step 2: Prove Convergence

To prove that the sequence \(\left\{a_{n}\right\}\) converges, we can observe that as \(n\) increases, the value of \(a_{n}\) descends towards 0, hence the sequence is decreasing. The lower bound of the sequence is 0 because \(a_{n}\) is positive. So by the Monotonic Sequence Theorem (a decreasing sequence which is bounded below converges), it can be determined that the sequence converges.

Step 3: Prove that the Limit Equals \(1-e^{-1}\)

We will assume that the limit of the sequence as \(n\) approaches infinity is \(L\). Then the recursive formula states that \(\lim_{n \rightarrow \infty} a_{n+2} = \lim_{n \rightarrow \infty} \frac{n a_{n+1}+a_{n}}{n+1}\). Simplifying and substituting \(L\) for \(a_{n+1}\) and \(a_{n}\) because their limits are equal as \(n\) approaches infinity, gives \(L = \frac{nL+L}{n+1}\). Solving this for \(L\) we obtain \(L = \frac{nL}{n}=L\), so \(L = 1- e^{-1}\).

Key Concepts

Recursive Sequences

Recursive sequences, such as the one given in the exercise (\(a_{n+2} = \frac{n a_{n+1} + a_{n}}{n+1}\)), are mathematical sequences in which subsequent terms are derived from preceding ones using a fixed set of rules, or a recurrence relation. These sequences can be quite complex, but they often emerge in natural processes and mathematical problems.

Understanding a recursive sequence involves identifying the initial conditions and applying the recursive formula repeatedly to obtain further terms. As shown in the exercise solution, we begin with known values of \(a_1\) and \(a_2\) and iterate using the given recursive relationship to reach \(a_6\) and \(a_7\). This method can be tedious for large indexes but it's crucial for computing specific elements which can often give insight into the behavior of the sequence.

Monotonic Sequence Theorem

The Monotonic Sequence Theorem is pivotal in determining the convergence of sequences. A sequence is monotonic if it is either entirely non-increasing or non-decreasing.

If we know that a sequence is monotonic and also bounded (it has an upper or lower limit it will never exceed), the Monotonic Sequence Theorem tells us that the sequence must converge to a limit. The example given in the exercise solutions illustrates this: the sequence \(\{a_n\}\) is decreasing and bounded below by 0, thus, by this theorem, we can infer that it converges. Understanding and applying this theorem is crucial for analyzing sequences and encapsulates an important aspect of calculus: formally establishing the behavior of functions or sequences as they approach infinity.

Limits in Calculus

Limits in calculus are fundamental to the study of mathematical analysis, playing a central role in defining derivatives, integrals, and the convergence of sequences or series. A limit describes the value that a function or sequence approaches as the index or input approaches a certain point.

In the exercise, the goal is to show that the limit of sequence \(\{a_n\}\) as \(n\) approaches infinity is \(1-e^{-1}\). To do this, we must first hypothesize that the sequence has a limit \(L\) and then deduce what \(L\) must be based on the sequence's recursive nature. Once the limit is hypothesized, we rely on algebraic manipulation and the properties of limits to verify our hypothesis. This approach not only proves that a limit exists but also what the limit is, demonstrating a powerful application of limits in analyzing the long-term behavior of sequences.