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(a) By completing the square, show that $ \int^{1/2}_0 \frac {dx}{x^2 - x + 1} = \frac {\pi}{3 \sqrt 3} $(b) By factoring $ x^3 + 1 $ as a sum of cubes, rewrite the integral in part (a). Then express $ 1/(x^3 + 1) $ as the sum of a power series and use it to prove the following formula for $ \pi: $ $ \pi = \frac {3 \sqrt 3 }{4} \sum_{n = 0}^{\infty} \frac {(-1)^n}{8^n} \left( \frac {2}{3n + 1} + \frac {1}{3n + 2} \right) $
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Calculus 2 / BC
Chapter 11
Infinite Sequences and Series
Section 9
Representations of Functions as Power Series
Sequences
Series
Campbell University
Harvey Mudd College
Baylor University
University of Michigan - Ann Arbor
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.
17:21
(a) By completing the squa…
01:35
(a) Make an appropriate $u…
01:58
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