Grandi's series | Parity (mathematics) | Divergent series
Guido Grandi (1671–1742) reportedly provided a simplistic account of the series in 1703. He noticed that inserting parentheses into 1 − 1 + 1 − 1 + · · · produced varying results: either or Grandi's explanation of this phenomenon became well known for its religious overtones: By putting parentheses into the expression 1 − 1 + 1 − 1 + · · · in different ways, I can, if I want, obtain 0 or 1. But then the idea of the creation ex nihilo is perfectly plausible. In fact, the series was not an idle subject for Grandi, and he didn't think it summed to either 0 or 1. Rather, like many mathematicians to follow, he thought the true value of the series was 1⁄2 for a variety of reasons. Grandi's mathematical treatment of 1 − 1 + 1 − 1 + · · · occurs in his 1703 book Quadratura circula et hyperbolae per infinitas hyperbolas geometrice exhibita. Broadly interpreting Grandi's work, he derived 1 − 1 + 1 − 1 + · · · = 1⁄2 through geometric reasoning connected with his investigation of the witch of Agnesi. Eighteenth-century mathematicians immediately translated and summarized his argument in analytical terms: for a generating circle with diameter a, the equation of the witch y = a3/(a2 + x2) has the series expansion and setting a = x = 1, one has 1 − 1 + 1 − 1 + · · · = 1⁄2. * According to Morris Kline, Grandi started with the binomial expansionand substituted x = 1 to get 1 − 1 + 1 − 1 + · · · = 1⁄2. Grandi "also argued that since the sum was both 0 and 1⁄2, he had proved that the world could be created out of nothing." Grandi offered a new explanation that 1 − 1 + 1 − 1 + · · · = 1⁄2 in 1710, both in the second edition of the Quadratura circula and in a new work, De Infinitis infinitorum, et infinite parvorum ordinibus disquisitio geometrica. Two brothers inherit a priceless gem from their father, whose will forbids them to sell it, so they agree that it will reside in each other's museums on alternating years. If this agreement lasts for all eternity between the brother's descendants, then the two families will each have half possession of the gem, even though it changes hands infinitely often. This argument was later criticized by Leibniz. The parable of the gem is the first of two additions to the discussion of the corollary that Grandi added to the second edition. The second repeats the link between the series and the creation of the universe by God: Sed inquies: aggregatum ex infinitis differentiis infinitarum ipsi DV æqualium, sive continuè, sive alternè sumptarum, est demum summa ex infinitis nullitatibus, seu 0, quomodo ergo quantitatem notabilem aggreget? At repono, eam Infiniti vim agnoscendam, ut etiam quod per se nullum est multiplicando, in aliquid commutet, sicuti finitam magnitudiné dividendo, in nullam degenerare cogit: unde per infinitam Dei Creatoris potentiam omnia ex nihlo facta, omniaque in nihilum redigi posse: neque adeò absurdum esse, quantitatem aliquam, ut ita dicam, creari per infinitam vel multiplicationem, vel additionem ipsius nihili, aut quodvis quantum infinita divisione, aut subductione in nihilum redigit. (Wikipedia).
Greek Mathematics: The Beginning of Greek Math & Greek Numerals
Welcome to the History of Greek Mathematics mini-series! This series is a short introduction to Math History as a subject and the some of the important theorems created in ancient Greece. You are watching the first video in the series. If this series interested you check out our blog for
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Who Were the Mongols? | Genghis Khan and The History of the Mongol Empire
Hear the story of one of the most violent empires that the world has ever seen—the Mongol Empire—beginning with Genghis (Chinggis) Khan, who claimed a mandate from heaven to rule the world. The rest is a story filled with murder, destruction, and the ultimate unification of Eurasia. This
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Drawn History: The 1930s | History
Learn more about the 1930s, a particularly tumultuous decade in world history that got its start with a bang - or, more accurately, a crash. Subscribe for more HISTORY: http://histv.co/SubscribeHistoryYT Newsletter: https://www.history.com/newsletter Website - http://www.history.com /pos
From playlist Drawn History: Illustrations & Animations of Historical Events | History
I Just Can't Believe what the Math is Telling Me
The Grandi's Series history is quite famous. The sum 1-1+1-1+1-1 to infinity should clearly diverge with no dispute. However, this 1-1+1-1+1-1 infinite series has a very controversial answer: 1-1+1-1 = 1/2 ?! How could this be? Here's my option on the Grandi's Series answer and a Grandi'
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Ancient Empire Builders | History Lists | History
Get an overview of the ancient leaders who oversaw the expansion and growth of civilizations around the world. "Newsletter: https://www.history.com/newsletter Website - http://www.history.com Facebook - https://www.facebook.com/History Twitter - https://twitter.com/history" HISTORY Topic
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One minus one plus one minus one - Numberphile
Discussing the brain-bending Grandi's Series and Thomson's Lamp - featuring Dr James Grime. More links & stuff in full description below ↓↓↓ A little bit of extra footage from the very end of this interview at: http://youtu.be/TgGi1ls1X-0 (on Brady's own channel) See our Zeno's Paradox v
From playlist James Grime on Numberphile
1-1+1-1+... Grandi series from a complex analysis angle
In this video I talk about switching the limits of an expression that amounts to the Grandi series - a series which is divergent in real and comlex analysis. I'm also ranting a bit on the History of Science and math StackExchange page and other communities.
From playlist Programming
History of the Holidays: History of Thanksgiving | History
Although Thanksgiving celebrations dated back to the first European settlements in America, it was not until the 1860s that Abraham Lincoln declared the last Thursday of November to be a national holiday. HISTORY®, now reaching more than 98 million homes, is the leading destination for aw
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Napoleon's Bloodless Coup | History
In 1799, Napoleon Bonaparte pulled off one of the greatest political coups in history, seizing control of the French government in a matter of days – without sacrificing a single life. #HistoryChannel Subscribe for more from HISTORY: http://histv.co/SubscribeHistoryYT Read more: http:/
From playlist Examine the Past | History
Ten Minute History - The Unification of Italy (Short Documentary)
Twitter: https://twitter.com/Tenminhistory Patreon: https://www.patreon.com/user?u=4973164 This episode of Ten Minute History (like a documentary, only shorter) covers Italy in the wake of the Napoleonic Wars and the tumultuous decades which followed it. The focus is on liberal revolution
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The third video in a series about Ramanujan.This one is about Ramanujan Summation. Here's the wikipedia page for further reading: https://en.wikipedia.org/wiki/Ramanujan_summation Euler-Maclaurin Formula https://en.wikipedia.org/wiki/Euler%E2%80%93Maclaurin_formula --------- Here is a
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The Roman Empire. Or Republic. Or...Which Was It?: Crash Course World History #10
In which John Green explores exactly when Rome went from being the Roman Republic to the Roman Empire. Here's a hint: it had something to do with Julius Caesar, but maybe less than you think. Find out how Caesar came to rule the empire, what led to him getting stabbed 23 times on the floor
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11. New Modes and Orders: Machiavelli's The Prince (chaps. 13-26)
Introduction to Political Philosophy (PLSC 114) The discussion of Machiavelli's politics continues in the context of his most famous work, The Prince. A reformer of the moral Christian and classical concepts of goodness and evil, Machiavelli proposes his own definitions of virtue and vi
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Thoughts on the 1-1+1-1+... Series - A Gentle Discussion About Analytic Continuation
We're going to mess around with some divergent series in this video, namely the Grandi Series, 1-1+1-1+... and the related series 1-2+3-4+... and I'll hopefully convince you that the sums that are associated with these divergent series make some sense when seen through the lens of calculus
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Today we examine this amazing, spine-backed Mammal-like Reptile or Synapsid! We will try to answer questions like: Did dimetrodon have fur? How did dimetrodon walk? And what is the deal with dimetrodon's back? Sources: http://www.bioone.org/doi/abs/10.3158/2158-5520-5.1.104 https://en.wik
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Solving the Basel Problem ... With a Divergent Series!
The Basel problem asks for the sum of the reciprocals of the square numbers, ζ(2), and was famously solved by Euler. In this video, we will find that a simple argument can be given for the solution of the Basel problem using a mysterious divergent series, which upon some simple operations
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Deconstructing History: New Orleans | History
How long does Mardi Gras last? How many beads get thrown every year? Learn the facts and figures behind the big festivities in the Big Easy. Subscribe for more from HISTORY on YouTube: http://histv.co/SubscribeHistoryYT Newsletter: https://www.history.com/newsletter Website - http://www.
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Italian Culture: The History of Tiramisù
The most beloved Italian dessert, at least in the opinion of Americans, is tiramisù. People just love this stuff! So when was it created, in Ancient Roman times? During the Renaissance? Nope! In the 1970s it turns out. Let's learn about Roberto Linguanotto and the invention of tiramisù now
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This Video Will Open Your Mind
Head to https://squarespace.com/brithemathguy to save 10% off your first purchase of a website or domain using code BRITHEMATHGUY The 1-2+3-4+5-6 series is a bit a mathematical wonder. On one hand, this 1-2+3-4+5-6 series diverges. On the other hand, there are some techniques to assign in
From playlist Series, Series, Series