Ep 17. The Joy of Infinite Series

The Joy of Mathematics is a television show that explores the beauty and complexity of mathematics. In season 1 episode 17, titled The Joy of Infinite Series, viewers are taken on a journey through the fascinating world of infinite series.

The episode begins with an explanation of what an infinite series is. Simply put, an infinite series is the sum of an infinite sequence of numbers. This may seem like a daunting concept, but the show presents it in a way that is easy to understand. Viewers are shown visual representations of infinite series and are introduced to some of the most famous examples, such as the geometric series and the harmonic series.

One of the most interesting parts of the episode is when the show delves into the history of infinite series. Viewers learn that the concept of an infinite series dates back to ancient Greece, with the mathematician Zeno of Elea famously proposing a paradox involving an infinite sum. The show then fast-forwards to the 17th century and takes a look at the contributions of mathematicians such as Isaac Newton and Gottfried Leibniz, who used infinite series to solve problems in calculus.

The episode also covers the topic of convergence, which refers to whether an infinite series adds up to a finite number or not. Viewers are shown how to determine whether an infinite series is convergent or divergent, and are introduced to criteria such as the ratio test and the integral test.

One of the most fascinating parts of the episode is when the show explores a real-life application of infinite series. Viewers are introduced to a Japanese art called "notan", which is based on the concept of dividing a square into smaller and smaller sections. The resulting pattern can be represented as an infinite series, and the show demonstrates how this can be used to create beautiful artwork.

Overall, The Joy of Infinite Series is an engaging and informative episode that provides viewers with a newfound appreciation for the beauty of mathematics. From the history of infinite series to the practical applications in art, this episode covers a wide range of topics that are sure to capture the interest of both math enthusiasts and casual viewers alike.