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In Exercise 10.2.53 it was shown that the length of the ellipse $ x = a \sin \theta, y = b \cos \theta, $ where $ a > b > 0, $ is
$ L = 4a \int^{\pi/2}_0 \sqrt {1 - e^2 \sin^2 \theta} $ $ d\theta $
where $ e = \sqrt {a^2 - b^2}/a $ is the eccentricity of the ellipse.
Expand the integrand as a binomial series and use the result of Exercise 7.1.50 to express $ L $ as a series in powers of the eccentricity up to the term in $ e^6. $
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Calculus 2 / BC
Chapter 11
Infinite Sequences and Series
Section 10
Taylor and Maclaurin Series
Sequences
Series
Campbell University
University of Michigan - Ann Arbor
University of Nottingham
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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The problem is in exercise 10.253. It shown that the length of the ellipse x is equal to sine theta y is equal to b times, cosine theta, why a is greater than b is greater than 0 is l is equal to 4 a times integral from 0 from from 0 to pi over 2, integral is square root of 1 minus x square times sine t square, where a is equal to square root of a square. Minus b, squared and divided by a is accentricity of the ellipse, extend it integrant as a binomial series and use the result of exercise 7.1 .50 to express l as a series of powers of the eccentricity up to the term in e to 6. So first we have 1 plus x to the power of 1. Half is equal to some and from 0 to infinity on half in x to m, so we half the integrant square root of 1 minus a square sine theta square is equal to sum, and from 0 to infinity on half n times negative, a square sine square To the power of n, which is equal to some and from 0 to infinity, while half n times negative 1 to n times e to 2 n times sine theta to the power 2, then the half l is equal to 4 a times. So the first term is integral 0 to pi over 2 times 1, the tea and plus 1 half times negative 1 times a square times integral from 0 to pi over 2 sine theta square d, theta plus 1 half times negative 1 half over 2 factorial times E to 4 times integral from 0 to pi over 2 sine t to the power 4 d, theta and plus 1 half times negative 1 half times negative 3, half over 3 factorial times negative 1 times e to 6 and times integral from 0 to pi. Over 2 sine theta to the power 6 d, theta and plus the the and by using the results of exercise 7.1 .50, we have integral, from 0 to pi over 2 sine theta to the power of a power of 2 n d. Theta is equal to 1 times 3 times 5 times dated times: 2 n minus 1 over 2 times 4 times the times 2 n times pi over 2 point. So behalf. This is equal to 4 a times pi over 2 minus pi over 8 times a square minus 3 pi over 128 times e to 4 minus 5 pi over 512 times e to 6 plus. Do.
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