Fun With Fourier Series

By using computers to do experimental manipulations on Fourier series, we construct additional series with interesting properties. For example, we construct several series whose sums remain unchanged when the nth term is multiplied by sin(n)/n. One series with this property is this classic series for pi/4: pi/4 = 1 - 1/3 + 1/5 ... = 1*(sin(1)/1) - (1/3)*(sin(3)/3) + (1/5)*(sin(5)/5).... Another example is sum (n = 1 to infinity) of (sin(n)/n) = sum (n = 1 to infinity) of (sin(n)/n)^2 = (pi - 1)/2. This material should be accessible to undergraduates. This paper also includes a Mathematica package that makes it easy to calculate and graph the Fourier series of many types of functions.

Summing the curious series of Kempner and Irwin

In 1914, Kempner proved that the series 1/1 + 1/2 + ... + 1/8 + 1/10 + 1/11 + ... + 1/18 + 1/20 + 1/21 + ... where the denominators are the positive integers that do not contain the digit 9, converges to a sum less than 90. The actual sum is about 22.92068. In 1916, Irwin proved that the sum of 1/n where n has at most a finite number of 9s is also a convergent series. We show how to compute sums of Irwins series to high precision. For example, the sum of the series 1/9 + 1/19 + 1/29 + 1/39 + 1/49 + ... where the denominators have exactly one 9, is about 23.04428708074784831968. Another example: the sum of 1/n where n has exactly 100 zeros is about 10 ln(10) + 1.007x10^-197 ~ 23.02585; note that the first, and largest, term in this series is the tiny 1/googol.

Fun With Very Large Numbers

We give an example of a formula involving the sinc function that holds for every N = 0, 1, 2, ..., up to about 10^102832732165, then fails for all larger N. We give another example that begins to fail after about N ~ exp(exp(exp(exp(exp(exp(e)))))). This number is larger than the Skewes numbers.

Experiments with zeta zeros and Perrons formula

Of what use are the zeros of the Riemann zeta function? We can use sums involving zeta zeros to count the primes up to $x$. Perrons formula leads to sums over zeta zeros that can count the squarefree integers up to $x$, or tally Eulers $phi$ function and other arithmetical functions. This is largely a presentation of experimental results.