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Prove Theorem 7.

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$\lim _{n \rightarrow \infty} f\left(a_{n}\right)=f(L)$

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 1

Sequences

Series

Missouri State University

Harvey Mudd College

University of Nottingham

Idaho State University

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.

both here, um, seven states that if the Limited and Ezel and efforts continue settle, then the limit. So this is the statement of the pharaoh. The limit of f of a end as n goes to infinity equals f f l. So the proof for this is very short weaken, right. The limit as n goes to infinity half of an peoples f of the limit is n goes to infinity of a n Why can we do this Since efforts continue Asado This is just the definition of continuity at a point l now on. On the other hand, looking at the inside, we know that the limit of Anna's cell So this justifies the final equation. So we started with the limit of f ve n. We ended up with a f of l. That was the claim that was to be proved. We have proved it. That's our final answer.

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