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Problem 42 Hard Difficulty

(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

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Section 9

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Series - Intro

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.

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Sequences - Intro

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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Video Transcript

part of a by completing the squire showed that into girl from zero to one half one over X squared minus X plus one X is equal to pi over three times three. So we're half into grew your own behalf Yaks over X squire months X plus one I'm completing the square. Have this's equal into girl From there on my half the night's over. Thanks, Linus Squire. Us two three over two square. So since is equal to one over you took three over two hams. Attendant X minus one half over two three over too. From zero to a half. This is control through over to three times. Hi. Over six, which is? They were to pi over three houses through to three. Hard to be by factoring Axe Cube plus one has a some ofthe cubes. Rewrite it into growing Impart a then express one number X cubed plus one. I'm the song off power Siri's and you need to prove falling form formula for hi Affairs to behalf one over. Excuse minus X off one. This is sick too. X plus one over X cube. What? This is it, Max. US one harms one over love plus x Q. We can write it. This is a new song from you're only twenty Negative. Want the part ofthe Ken Ham's X to the power of three hands? So this is a call to some from zero to infinity Makes you want to end comes axe to cover three plus one plus thanks to the power of three, then the integral from zero to one half and to grow from zero to one half. This is the sequence of the integral from zero to one half the ex which is in control, some from so your own between. Fine aunty that you want in Homs Axe to department three plus two over three can trust too US three plus one over three hundred plus one from zero to Is this axe to power? Billion possible But this is a call to some throne. So you're off to infinity. Like to want to bend over and to end times another floor over three hand plus two us over three can us lot. So from part a half the skin to grow is also equal to pi over three town streets of three So we have pie over three two three Go to Som. From zero to infinity. Like to want and over it to end here. Provided this one force here are forced hams, Some from zero to infinity. Makes you want to end over end to end. Hams is one over three plus two plus two over there. What then be much fly three times. Spirit of three for each side, half pie. His secret to this function.

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Series - Intro

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Video Thumbnail

02:28

Sequences - Intro

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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