7/04/2020

Why Is 1 + 2 + 4 + 8 + … = -1



The Powers of Two: Why Is 1 + 2 + 4 + 8 + … = -1

On calculating infinite divergent series sums


Jul 1 · 5 min read




The Short Answer

To satisfy you’re curiosity and save you from the mathematical jargon, the simple explanation is just:
x = 1 + 2 + 4 + 8 + …
x = 1+ (2 + 4 + 8 + …)
x = 1+ 2(1 + 2+ 4 + 8…)
x = 1+ 2x
x = -1
Nonetheless, if you would join me in the scenic route, perhaps we can stumble upon a few more answers and — even better — more insightful questions.

Divergence vs Convergence

A convergent series is one whose sum approaches a limit.
For instance, the convergent series 1/2 + 1/4 + 1/8 + 1/16 + … clearly approaches a certain limit, which is 1, as made apparent by the geometric illustration below.


Partial Sums

We can also differentiate divergence and convergence by their partial sums. As the name suggests, a partial sum is a sum of a part of a series. We can express the partial sum of the first n terms of 1/2 + 1/4 + 1/8 … using the formula for a geometric series.


Applying that formula, we see that the partial sums of a convergent series seem to approach one, which is made even more evident by graphing it.

the graph of convergent partial sums approach one

In the case of divergence, however, the partial sums do not approach a value but extends to infinity

the graph of divergent partial sums go on infinitely

‘Rules’ For Calculating Infinite Divergent Series

Although doing so is comparatively less straightforward, it is still possible to get the sum of a divergent series — as long as we follow a few rules.

Regularity

A summation method is regular if the sum it gives for a convergent series is the limit of its partial sums


Linearity

To be linear, sums must be distributable and factorable


In a linearity, terms of a summation of equal length can be grouped.


Stability

Stability is present when terms can be “extracted” from a summation


Applications to power of two

x = 1 + 2 + 4 + 8 + …
(1) x = 1+ (2 + 4 + 8 + …)
(2) x = 1+ 2(1 + 2+ 4 + 8…)
x = 1+ 2x
x = -1
(1) x = 1+ (2 + 4 + 8 + …)


Hence we can say that the summation method is stable.
(2) x = 1+ 2(1 + 2+ 4 + 8…)


And finally, this shows that the series is linear.

The Long Answer

So given this, we can say that 1 + 2 + 4 + 8 + … is not completely -1, because the summation method we used — although linear and stable — is not totally regular. Totally regular summations would say infinity is an answer.
What’s the real answer? Depends which requirement you prioritize.
But it’s not always about finding the answer, is it? Just grappling with infinite divergent series can lead us to some interesting insights, and even — as Leonhard Euler has shown us — to profound discoveries on mathematics as a whole.
All it takes is time and patience.

Source

Sums Of Divergent Series. Brilliant.org. Retrieved July 1, 2020, from https://brilliant.org/wiki/sums-of-divergent-series/


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Living the complexity of the unknown and the uncertainty of the everyday | www.joaquindecastro.gq

7/01/2020

Kawanable Kyōsai’s Night Parade of One Hundred Demons (1890)


Kawanable Kyōsai’s Night Parade of One Hundred Demons (1890)

Kawanabe Kyōsai (1831–1889), aka “The Demon of Painting”, composed this book of woodblock illustrations toward the end of a life that had begun during the Edo period, when Japan was still a feudal country, and ended in the midst of the Meiji period, when the country was transforming into a modern state.
Kyōsai was by all accounts the bad boy artist of his era. Considered both Japan’s first political caricaturist and one of the first authors of a manga magazine (Eshunbun Nipponchi), Kyōsai was arrested by the shogunate three times for his commitment to free expression. Also, he made no secret of his love for sake.
The Night Parade of One Hundred Demons (Hyakki Yagyō) is a thousand-plus-year-old Japanese folkloric tradition, in which a series of demons parades — or explodes — into the ordinary human world.
Kyōsai’s version was, according to the Metropolitan Museum of Art which houses the book, one of the artist’s most popular volumes, offering “a spectacular visual encyclopedia of supernatural creatures of premodern Japanese folklore”. (To see more examples of such supernatural creatures, also see our post on this Edo-era scroll.)
One can see why it was so popular. Narratively, it paves the way for the fantastic parade with two woodblocks: the first depicts a group of adults and children gathered around a coal fire to hear ghosts stories, the second a man (probably Kyōsai) setting down his calligraphy brush and extinguishing the lamp in preparation for the night in which the demons will appear.
kyosai one hundred demons
Adults and children huddle around a brazier, or coal fire, to hear ghost stories.
kyosai one hundred demons
A man, perhaps the artist himself, has set down his calligraphy brush and reaches to extinguish a lamp. Once darkness falls, the demons will appear.
The illustrations of the demons themselves are appropriately terrifying. Skeletal soldiers riding a human-headed horse; a frog-like demon dominating a badger; furry-headed demons and naked demons that look like the stuff of Jim Henson’s darkest nightmares — all parade across Kyōsai’s pages.
Each double-page of the book is arranged in such a way as to join up with the next, as though a continuous scroll is divided over the pages of a book. Though be aware that, of course, the book is bound on the right and so runs counter to the usual left-to-right of English-language books, and so also counter to how our gallery below is set up to display!
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