The spooky-smart mathematician Srinivasa Ramanujan came up with this formula around 1913. Why is it true?
I don't know, let's see...
In 1735, a young fellow named Euler stunned the world by cracking a famous puzzle that had been unsolved for almost a century: the Basel problem. The problem was to sum the reciprocals of perfect squares:
1/1² + 1/2² + 1/3² + 1/4² + 1/5² ... = ???
Euler showed that the answer was π²/6:
1/1² + 1/2² + 1/3² + 1/4² + 1/5² ... = π²/6
He also showed you could rewrite this sum as a product over primes:
1/1² + 1/2² + 1/3² + 1/4² + 1/5² ... =
(2²/(2² - 1)) (3²/(3² - 1)) (5²/(5² - 1)) (7²/(7² - 1)) ...
That's actually the easy part: it's a cute trick called the Euler product formula.
So we know
(2²/(2² - 1)) (3²/(3² - 1)) (5²/(5² - 1)) (7²/(7² - 1)) ... = π²/6
If you think about it, Ramanujan's formula is saying that
(2²/(2² + 1)) (3²/(3² + 1)) (5²/(5² + 1)) (7²/(7² + 1)) ...
is 2/5 as big. So, proving it is the same as showing
(2²/(2² + 1)) (3²/(3² + 1)) (5²/(5² + 1)) (7²/(7² + 1)) ... = π²/15
Maybe the next step is to use the same idea as the Euler product formula. I think this gives
(2²/(2² + 1)) (3²/(3² + 1)) (5²/(5² + 1)) (7²/(7² + 1)) ... =
1/1² - 1/2² - 1/3² + 1/4² + 1/5² - 1/6² + 1/7² + ...
where the signs at right follow a fancy pattern: we get 1/n² whenever n is the product of an even number of primes, and -1/n² when n is the product of an odd number of primes. For example, 4 = 2 x 2 is the product of an even number of primes, so we get 1/4².
So I'm left wanting to know why this strange sum
1/1² - 1/2² - 1/3² + 1/4² + 1/5² - 1/6² + 1/7² + ...
equals π²/15. Ramanujan, dead since 1920, is still messing with my mind!
The formula is supposed to be in here:
• Srinivasa Ramanujan, Modular equations and approximations to π, Quart. J. Pure. Appl. Math. 45 (1913-1914), 350-372. Also available at ://ramanujan.sirinudi.org/Volumes/published/ram06.pdf.
But I don't see it!
Here you can see how Euler solved the Basel problem:
It's a great example of his brilliant tactics, many of which were far from rigorous by today's standards... but can be made rigorous.
You can now make your own cyborg roach for just $100. Just buy this kit developed by the company Backyard Brains:
Are you a teacher or parent that wants to teach a student about advanced neurotechnologies? You are in luck! After 3 long years of R&D, the RoboRoach is now ready for its grand release! We are excited to announce the world's first commercially available cyborg! With our RoboRoach you can briefly wirelessly control the left/right movement of a cockroach by microstimulation of the antenna nerves. The RoboRoach is a great way to learn about neural microstimulation, learning, and electronics!
We are recently ran a successfully-funded kickstarter campaign to fund the release of our new RoboRoach! The hardware and firmware development are complete and we are now shipping!
The RoboRoach "backpack" weighs 4.4 grams with the battery, and each battery will last over a month! Following a brief surgery you perform on the cockroach to attach the silver electrodes to the antenna, you can attach the backpack to the roach and control its movement for a few minutes before the cockroach adapts. When you return the cockroach to its cage for ~20 minutes, he "forgets" and the stimulation works again. Once you receive your RoboRoach in the mail, follow our online surgery instructions and videos and you will soon be on your way to becoming an expert in neural interfaces. After about 2-7 days, the stimulation stops working altogether, so you can clip the wires and retire the cockroach to your breeder colony to spend the rest of its days making more cockroaches for you and eating your lettuce.
1x Free iOS or Android 4.3+ application for remote control
1x Bluetooth Roboroach backpack control unit
1x 1632 RoboRoach Battery
3x Electrode Sets (to implant 3 Roaches)
View our RoboRoach Ethics Statement
Backyard Brains has developed ethical guidelines for all our products. You can read more in our statement regarding our use of insect for experiments at:
I feel ethical qualms about taking away the autonomy of an animal this way, and their ethics statement doesn't really address that. This is the closest they come:
Criticism: Modifying a living creature to make a toy is wrong.
The RoboRoach circuit is not a toy. This new bluetooth version is a powerful low-cost tool for studying neural circuits, allowing for students to make discoveries. High school students in New York, for example, have discovered random stimulation causes much slower adaptation times. We have scientist and high school educator colleagues who are mentoring students in novel behavioral experiments using the RoboRoach circuit. Some highlights will be posted on our website soon.
By focusing on the question of whether the RoboRoach is a "toy", they dodge the harder question of when it's okay to override the nervous system of an animal and make it do what you want. Perhaps feeling a bit nervous about this, some of the cyborg roach developers say they want to use it as a "rescue robot" that can crawl around and hear people trapped under collapsed buildings. I think most people would say this is okay, at least if it actually works.
For a critical view on the ethics, see:
For more on how to actually make a RoboRoach, go here:
- GBHS, AtturMaths, 1990 - 1997
- St. Joseph's TrichyPhysics, 1997 - 2000
- Bharathidasan UniversityComputer Applications, 2000 - 2003
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