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Use the series in Example 13(b) to evaluate

$ \lim_ {x \to 0} \frac {\tan x - x}{x^3} $

We found this limit in Example 4.4.4 using 1' Hospital's Rule three times. Which method do you prefer?

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Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 10

Taylor and Maclaurin Series

Sequences

Series

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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.

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Use the series in Example …

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Use series to evaluate the…

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Use Taylor series to evalu…

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02:00

use the serious, in example, her team party to be valid. So this limits. Okay, so lame. Ex too. It goes to zero tendon X minus X over. Execute. We can expand tender necks. I'm somebody. So it's going to be limb x zero. Explore s X cubed over three, plus some hair of the term of Kyra, Turn of X cube, Mona's X over X cube. And it becomes who the X goes to zero. So one third, one third times, one third plus some higher in terms of us, Hugh, over X cube. And finally it becomes to one third. And this is our limits for this for this question, all right.

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