Forever?

In my new tutoring job, I am working with Calculus students who are not doing as well as they (or their parents) would like. Our most recent topic has been infinite series and I am reminded of how much I loved teaching infinite series. The concepts involved in series are some of the least intuitive in mathematics, but some of the results are the most amazing in mathematics as well. A series of numbers is a sum: 1 + 1/2 + 1/4 + 1/8 + ... We are especially interested in sums that converge. If you look at the sum of the series in the last line after each new term is added, you see a pattern. After the first term, the sum is 1; after you add the second term the sum is 1 1/2. After the third term is added, 1 3/4 ( 1 + 1/2 + 1/4); 1 7/8 after 1/8 is added, and so on. The sum keeps getting closer and closer to 2, but will never quite reach it. Mathematically we say the sum after infinitely many terms is 2.
What makes an infinite series interesting is that functions have infinite series representations. I don't know if you gave it any thought in your high school trig class, but when you typed sin (1) into your calculator, your calculator did a series to figure it out. My students always assumed that the calculator had a table to look up the values, but if you have done anything with computers, you know how much memory tables take up, A calculator would need to have tables that would include sin 1, sin 1.1, sin 1.11, sin 1.111, and so forth. That's a lot of values to remember. More importantly,  there is the question of where do the values in the table come from.
Luckily the sine function has an infinite series:

Those exclamation points are factorials: 3! means 3 X 2 X 1 = 6;   5! means 5X4X3X2X1 = 120;   7! means 7 X 6 X 5 X 4 X 3 X 2 X 1= 5040, and so on.
So sin 1 = 1 - 1/6 + 1/120 - 1/5040 + 1/362880 - ....  Your calculator just keeps adding terms following this pattern until it doesn't see any changes in the first twelve or so decimal places. Then it shows the answer as a decimal rounded to 8 decimal places (depending on your calculator).  Instead of a table with lots and lots of values, it just needs to remember a formula.

If you have invented a fast new super-computer and you would like to compare its speed to the old ones, one way to do it is to use an infinite series to find pi to lots of decimal places. Yes, that's how someone figures out the first fifty digits of pi:

                          

A really easy series to use is  pi =  4/1  -  4/3  +  4/5  -  4/7  +  4/9  -  4/11  +  ..., but this series converges pretty slowly, so it takes a lot of terms to get an accurate estimate for pi (90 terms just to get to 3.14).  So over the years mathematicians have come up with more complicated infinite series to compute pi.   

Using series and a little guidance, my calculus students would generate a surprising formula attributed to Leonhard Euler (pronounced Oiler), a highly regarded Swiss mathematician of the middle 1700's. A few days ahead of this, I would ask my students to list what they thought were the most important numbers that lay at the foundation of mathematics. Every year, the group would come up with the same five numbers 0, 1, pi, i (the imaginary unit - the square root of -1),  and e (about 2.718 - the base of natural logarithms - a huge topic in calculus). Then we would derive Euler's formula using infinite series: 

e ^iπ  +  1  =  0;  all five of the numbers they picked in one simple formula.  Isn't math amazing?