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(a) Let $ a_1 =a, a_2 = f(a), a_3 = f(a_2) = f( f(a)), . . . , a_{n + 1} = f(a_n), $ where $ f $ is a continuous function. If $ lim_{n \to\infty} a_n = L, $ show that $ f(L) = L. $ (b) Illustrate part (a) by taking $ f(x) = \cos x, a = 1, $ and estimating the value of $ L $ to five decimal places.

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Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 1

Sequences

Series

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University of Michigan - Ann Arbor

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Boston College

Lectures

01:59

In mathematics, a series is, informally speaking, the sum of the terms of an infinite sequence. The sum of a finite sequence of real numbers is called a finite series. The sum of an infinite sequence of real numbers may or may not have a well-defined sum, and may or may not be equal to the limit of the sequence, if it exists. The study of the sums of infinite sequences is a major area in mathematics known as analysis.

02:28

In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed. Like a set, it contains members (also called elements, or terms). The number of elements (possibly infinite) is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence. Formally, a sequence can be defined as a function whose domain is either the set of the natural numbers (for infinite sequences) or the set of the first "n" natural numbers (for a finite sequence). A sequence can be thought of as a list of elements with a particular order. Sequences are useful in a number of mathematical disciplines for studying functions, spaces, and other mathematical structures using the convergence properties of sequences. In particular, sequences are the basis for series, which are important in differential equations and analysis. Sequences are also of interest in their own right and can be studied as patterns or puzzles, such as in the study of prime numbers.

02:24

$\begin{array}{l}{\text { …

02:16

(a) Let $a_{1}=a, a_{2}=f(…

03:17

03:34

Let $f$ be the function $f…

So notice our sequence here we're starting off. Our first term is a constant and then the general term is given by Ryker. Shin or F is continuous. So we're trying to show a f of l equals l. So let's start with the left hand side Now, using the definition of El, I can rewrite Ellis this limit over here and now. Here's the key step. Using continuity of f, we can take the limit from inside and put it outside. So this is key here. This is given information that we're using f his continuous allows youto pull the limit outside. Now half of a n buy up. This this expression up here we know that it's just an plus one. And if you notice over here the limit of a gazelle, if you add one to the end, it's and it's still going to infinity. So this expression is also equal to help. And that's what we wanted to show f of l equals l and part B. They're giving you a constant A and a specific function f and we want to find the value l In our case, we could just go ahead and put a plus one. And this will be limit as n goes to Infinity Co signed to the end power of one. And so at this point, I would just go into a calculator, preferably a scientific. You have co sign of one and then insert that is co sign of the previous answer co sign of the previous answer and so on. And you want to do this until the first five decimal places stopped moving and the first time that should happen. The decimal is point seven three nine o nine. So this is a estimation of l two five decimal places, and that completes part B.

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