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Use series to evaluate the limit.

$ \lim_{x \to 0} \frac {\sqrt {1 + x} - 1 - \frac {1}{2} x }{x^2} $

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$-\frac{1}{8}$

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 10

Taylor and Maclaurin Series

Sequences

Series

Harvey Mudd College

University of Michigan - Ann Arbor

University of Nottingham

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.

01:17

Use series to evaluate the…

02:08

01:39

02:39

The problem is use, series to evaluate and limit limit x goes to 0 square root of 1 plus x, minus 1 minus 1 half times x over x squared. So we have, this, limit, is equal to limit x goes to 0 and we can. What root of 1 plus x has a series 1 plus 1 half x minus x, squared over 8 plus 1 over 16 x, cube plus to so here. We use the formula 1 plus x to the power of 1. Half is equal to series and from 0 to infinity, when half in times x to power and then minus 1 minus 1 half x over x square. This is equal to limit x, goes to 0 negative x, squared over 8 plus 116 n over 16 times x. Cube plus told over x coined so this is equal to negative 1 o 8.

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