(a) Show that if limx →∞ a2 n=L and limn →n a2 n+1=L, then...

(a) Show that if $\lim _{x \rightarrow \infty} a_{2 n}=L$ and $\lim _{n \rightarrow n} a_{2 n+1}=L$, then $\left\{a_{n}\right\}$ is convergent and $\lim _{n \rightarrow=} a_{n}=L$. (b) If $a_{1}=1$ and $$a_{n+1}=1+\frac{1}{1+a_{n}}$$ find the first eight terms of the sequence $\left\{a_{n}\right\}$. Then use part (a) to show that $\lim _{n \rightarrow \infty} a_{n}=\sqrt{2}$. This gives the continued fraction expansion $$\sqrt{2}=1+\frac{1}{2+\frac{1}{2+\ldots}}$$

(a) Show that if $\lim _{x \rightarrow \infty} a_{2 n}=L$ and $\lim _{n \rightarrow n} a_{2 n+1}=L$,
then $\left\{a_{n}\right\}$ is convergent and $\lim _{n \rightarrow=} a_{n}=L$.
(b) If $a_{1}=1$ and
$$a_{n+1}=1+\frac{1}{1+a_{n}}$$
find the first eight terms of the sequence $\left\{a_{n}\right\}$. Then use part (a) to show that $\lim _{n \rightarrow \infty} a_{n}=\sqrt{2}$. This gives the continued fraction expansion
$$\sqrt{2}=1+\frac{1}{2+\frac{1}{2+\ldots}}$$

Key Concepts

Continued Fractions

Continued fractions are expressions for numbers as the sum of an integer and the reciprocal of another number, which is itself represented in the same way ad infinitum. They provide an alternative representation of real numbers, particularly irrational numbers, and reveal interesting patterns in their structure. Continued fractions are valuable in number theory and approximation theory, offering insights into the irrationality and best rational approximations of numbers.

Fixed Point Analysis

Fixed point analysis involves finding a value that remains the same under a given function, often used to determine the limit of a recursive sequence. For sequences defined by the recurrence a_{n+1} = f(a_n), if the sequence converges to some limit L, then L must satisfy the equation f(L) = L. This self-referential condition is central to solving for limits in recursive sequences and is deeply connected to concepts in dynamical systems.

Recurrence Relations

Recurrence relations define sequences by expressing each term as a function of one or more previous terms. This creates a recursive structure, enabling the construction of the entire sequence based on initial value(s) and a recursion formula. Understanding recurrence relations is essential in numerical methods, dynamic programming, and the study of iterative algorithms.

Subsequence Convergence

Subsequence convergence is the study of the limits of subsets of the index set of a sequence, such as the even-indexed or odd-indexed elements. A crucial result in analysis is that if two subsequences cover all elements and both converge to the same limit, then the whole sequence must also converge to that limit. This principle is often used as a tool for analyzing more complex sequences by breaking them into simpler parts.

Sequence Convergence

Sequence convergence is the concept in mathematical analysis where a sequence approaches a specific value (limit) as the index goes to infinity. When a sequence converges, for any arbitrarily small positive number, there exists a point beyond which all terms of the sequence are within that distance from the limit. This is a fundamental idea in calculus and analysis, ensuring that the behavior of a sequence stabilizes near some number.

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