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An Application of Limits: Archimedes' Infinite Sum

Author:Greg Petrics
When we studied limits, we didn't get to see many applications of them. The reason was that I wanted to get us onto derivatives as quickly as possible. That said, now that we're at the end of the book, it might be fun to go back and take a look at famous example of a limit. In fact, this was one of the first limits that was ever calculated. The sum

never ends. The next term in the sum is always 1/4 of the term before it. So the next term would be 1/1024. Because this process is non-terminating, a limit is required to calculate it. A first guess is that this sum continues upwards to increase on up to infinity. However, if you get out your calculator (TRY IT), you'll notice that it doesn't get infinitely large. In fact, it doesn't even seem to get past 0.4! This infinite sum is an example of what's known as an infinite series. The term "infinite series" is just a fancy way of saying "infinite sum". This particular infinite series was one of the first to be calculated in the history of mathematics. Archimedes calculated it circa 250 BC. Archimedes noticed a very clever thing: if you add THREE of these infinite series together, the result is 1. He came to understand this because of a very clever visualization of the sum. The applet below helps you see what he saw in a dynamic way

And this is an excellent place to conclude Season 1 of Calculus for the People. With this activity, we've now seen applications of limits, derivatives and integrals. You are a bona-fide calculus understander now. Congrats. Furthermore, this is a great place to invite you to read Season 2 of Calculus for the People when it's out. The concept of an infinite series from this applet is not just an archaic oddity that only showed up 2000 years ago. To the contrary, series are powerful tools for doing important calculations that are used in almost all of the software that you use today. We'll study them in great detail in Season 2. Be on the lookout for that book in Fall 2020. In the meantime, feel free to email me at greg [dot] metrics [at] northernvermont [dot] edu, but replace the "m" with a "p" in "metrics". You can also leave anonymous feedback here. I'd love to know what you think! Thank you for reading!

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