Recall the formula we learned for the sine function in terms of an infinite product:

Thus:

.

Now, let us compare this to the Taylor series for sine:

Thus

and so

.

Now, if you consider the expansion of the infinite product , we see that all *x*^{2} terms arise from multiplying one term from the right of one parenthesis, and the ones from the rest of the parentheses: thus, we have terms etc.

Thus we have the coefficient of *x*^{2} as , but via the Taylor series, the coefficient is , thus telling us .

Note that this sum is , where is the Riemann zeta function introduced in this previous post. The problem of finding the exact value of this series and proving it was known as the Basel problem, and was first solved by Euler at age 28.

Now, let us consider the *x*^{4} terms: here, the result from the product formula are terms of the form , with *k* and *l* distinct, and only one such for each pair of distinct numbers. Thus, the coefficient is

, and by examining the Taylor series, we have

.

Consider: . First, we see that this sum is:

.

However, we can also break up the sum into three parts: , , and :

As the sum for is the same as for . We already found the value for these, and so:

.

And thus, we have the exact value for .

In fact, we can find the exact value for the zeta function at any positive even integer. More specifically, , where the are the Bernoulli numbers, which I have mentioned earlier in the context of Faulhaber’s Formula.

## Archive for June, 2008

### Monday Math 26

June 30, 2008### Worrying

June 28, 2008Mike the Mad Biologist reports on some problems at the Center for Disease Control’s Q fever lab, a Biosafety Level 3 lab. Most notably, it appears the containment door is sealed with *duct tape*!

### Medicare

June 28, 2008Scott Gottlieb reminds us, in the *Wall Street Journal*, “What’s at Stake in the Medicare Showdown.”

### Warnings

June 28, 2008### Four Stars

June 28, 2008President Bush has nominated Lt. Gen. Ann E. Dunwoody to serve as head of the United States Army Materiel Command. This is significant as, if confirmed by the Senate, General Dunwoody would become the first female four-star general in U.S. history. The usual path to that rank is via combat positions, from which women are still barred.

### Drug Use and Lung Damage

June 28, 2008DrugMonkey uses a recent celebrity example to discuss studies of the potentially severe consequences for lung health of the smoking of crack cocaine.

### Molecular Clutch

June 28, 2008ScienceDaily discusses the protein “clutch” in the flagellum of *Bacillus subtilis*.

The action of the protein they discovered, EpsE, is very similar to that of a car clutch. In cars, the clutch controls whether a car’s engine is connected to the parts that spin its wheels. With the engine and gears disengaged from each other, the car may continue to move, but only because of its prior momentum; the wheels are no longer powered.

EpsE is thought to “sit down,” as Kearns describes it, on the flagellum’s rotor, a donut-shaped structure at the base of the flagellum. EpsE’s interaction with a rotor protein called FliG causes a shape change in the rotor that disengages it from the flagellum’s proton-powered engine.

### For those of you stuck in the 80s…

June 28, 2008…comes the *Knight Rider* satellite navigation system, complete with the voice of William Daniels.