Infinite Series

In our infinite series class (Winter 2020), we will investigate a range of techniques for evaluating infinite series in closed form. Typically, students only learn how to evaluate a very small number of infinite series, such as geometric series and telescoping series. However, there are many fascinating approaches available for evaluating other types of infinite series.

One of the most famous infinite series is the sum $\sum_{n=1}^\infty \frac{1}{n^2}$. Evaluating this series by showing it sums to $\frac{\pi^2}{6}$ was one of Euler’s early triumphs, and we’ll see how to do it. A related sum, $\sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)^3}$, can also be evaluated, this time using techniques from Fourier series, so this will be a perfect time to introduce Fourier series.

Another famous series, which we will evaluate if time permits, is Ramanujan’s fast-converging formula for $\pi$: $\frac{1}{\pi}=\frac{2\sqrt{2}}{9801}\sum_{n=0}^\infty \frac{(4n)!(1103+26390n)}{(n!)^4 396^{4n}}$. This one is far more difficult and relates to deep results in number theory.

We will also look at related topics, such as infinite products, continued fractions, nested radicals, and so forth.

Calculus is required for this class.