In 1689 Jakob Bernoulli deduced that the series

1. Show that

   ¥ å n = 1  2 n(n+1) = 2.

   (Hint: Find a partial fraction decomposition and rewrite the series as a telescoping series.)
2. By comparing terms, argue that

   ¥ å n = 1  1 n2 < ¥ å n = 1  2 n(n+1) ,

   and thereby deduce that the first series converges to a number less than 2.
3. (Enter Leonhard Euler) Find the power series, centered at x = 0, for f(x) = sin(x).
4. Now consider the function F(x) = sin(x) / x and let's just say F(0) = 1 so that F is defined and continuous for all real numbers. Explain why we can say this.
5. Use your power series for f to find a power series for F.
6. Euler simply thought of the power series as an infinite polynomial. Since F(0) = 1 and F(kp) = 0 for any integer k, Euler concluded that

   F(x) = æ è 1 - x p ö ø æ è 1 - x -p ö ø æ è 1 - x 2p ö ø æ è 1- x -2p ö ø ¼

   (1)

   Explain why this seems to make sense.
7. Show that the equality above amounts to

   1 - x2 3! + x4 5! - x6 7! + ¼ = æ è 1 - x2 p2 ö ø æ è 1 - x2 4p2 ö ø æ è 1- x2 9p2 ö ø ¼

   (2)

8. Now we'll focus on the term involving x². First, on the left hand side of equation (2), notice that the coefficient of x² is -¹/₆. Now for the right hand side! Let's use Mathematica to determine the coefficient of x².
   1. We'll need to ``trick'' Mathematica so that it doesn't carry out certain computations. Enter a:={"1","2","3","4","5","6","7","8","9"} (We'll make Mathematica think these integers are text strings.)
   2. Enter b=Product[1-x^2/(Pi^2 a[[k]]^2), {k,1,9}]
   3. Find the coefficient of x² in the product by entering c=Coefficient[b, x^2]
   4. Now simplify. Simplify[c]
   5. What series forms the coefficient of x²?
9. Equate the coefficients of x² on the left and right hand sides of (2). What is the sum of the series

   ¥ å n = 1  1 n2 ?

10. (Extra Credit) Evaluate F and the right hand side of (2) at x = p/2. Obtain an infinite product whose value is p/2.