We began our very last class meeting with information about the final exam, including review sessions and office hours.

ndash History: the Basel problem
This problem was stated by Pietro Mengoli in 1644, and was solved by Euler almost a century later, in 1735. There is some question about the validity of his solution of the problem, and I'll address that later. The problem is to find some "analytical expression" or precise value for ∑n=1∞1/n2. The Wikipedia article discusses the problem and supplies several solutions. The problem is notorious, and has attracted a large number of methods of solution. Robin Chapman, an English mathematician, has assembled fourteen different proofs. You can also look at the very large number of references here.

Leonhard Euler
Euler was one of the greatest known mathematicians, and almost certainly the most prolific, writing incredible amounts of papers and books (including, as I mentioned, the first systematic calculus book). Here is a quote from another source:

Euler devises some clever approximations to get the {sum of the series] to six decimal places, 1.644934. Euler notes that to achieve such accuracy using direct calculation requires more than 1000 terms, so his estimate of the solution to the Basel problem was far more accurate than any available to his competitors. He later improved his estimate to 17 decimal places. Moreover, Euler was a genius at arithmetic, so he probably recognized this value as what will turn out to be the exact solution to the Basel problem, &Pi2/6. Euler didn't share this with the world, so he had a valuable 6 advantage as he raced to solve the problem; he knew the answer.
It is amazing to me that Euler could have done this, but he was quite remarkable. These days, smart people have constructing a web page called the Inverse Symbolic Calculator which can help people less gifted than Euler to recognize numbers. For example, when I enter 1.644934 in the ISC search bar one of the first suggestions I get is &zeta(2), which is a standard notation for ∑n=1∞1/n2. The ISC and the previously mentioned Encyclopedia of Integer Sequences are wonderful sources of amazing information.

The Vatter problems
By an amazing coincidence, Dr. Vincent Vatter, a former Rutgers math graduate student now at Dartmouth, visited me the morning before today's class. He asked me what I was teaching and when I told him today's topic he told me that the topic was a collection of problems in the calculus book, still in manuscript form. He and I printed the page containing the problems and then the class and I solved the problems. Here they are.

38. Derive the Maclaurin sderies for f(x)=sin(sqrt(x))/sqrt(x).

39. Find all the roots of f(x)=sin(sqrt(x))/sqrt(x), i.e., values of x for which f(x)=0.

40. If a polynomial of degree 3 has roots r1, r2, and r3, then it is given by p(x)-c(x-r1)(x-r2)(x-r3) for some constant c. By expanding this product, verify that
[1/r1}+[1/r2]+[1/r3]=- {coefficient of x}/{constant term}.
This fact is true for polynomials of any degree.)

41. Assume, as Euler did, that Exercise 40 holds for infinite series to show that
[1/r1}+[1/r2]+[1/r3]+...=1/6
where r1, r2, r3, ... are the roots from Exercise 39.
Conclude that ∑n=1∞1/n2=Π2/6.

42. The function f(x)=2-{1/(1-x)} has a single root, x=1/2. Derive its Maclaurin series and conclude that, contrary to Euler's assumption, Exercise 40 cannot be applied to infinite series.

I think the problems were mostly straightforward (people remembered in 42 to start with a geometric series).

I mentioned that Euler used similar techniques to find explicit values of ∑n=1∞1/nEVEN, where EVEN is any even integer (the values have the form (rational number) multiplying ΠEVEN). There is no known nice form for the sum of ∑n=1∞1/n3, even though a lot of people have thought about it for a long time.

So it was wrong, wasn't it?
So the proof quoted was certainly incorrect. This is interesting. But the historical source I quoted earlier noted that Euler gave at least three other solutions of the Basel Problem, and the "proof" we just discussed is the one most remembered. Several of Euler's other proofs use techniques familiar to all 152 students, but the proofs are definitely much more intricate.

Further, Euler was sort of correct although power series are not "exactly" like long polynomials. There are ways of relating the roots of a power series to the coefficients of the power series. About 150 years later, another mathematician named Weierstrass fixed up Euler's proof using a factorization theorem for power series. One of the key assumptions needed would rule out an example like the one given in problem 42 above. (Weierstrass, a great and famous mathematician, is less well known for being one of the lecturer's mathematical ancestors.

Rearranging a series
There are other strange things that happen with series. Many of these were known to Euler, and he mostly disregarded them. For example, we know that the Alternating Harmonic Series,
1-1/2+1/3-1/4+1/5-1/6+1/7...
converges since it satisfies all of the hypotheses of the Alternating Series Test. Call its sum, S (actually its sum is ln(2) but we don't need to know that here. Now let's rearrange it.

Write a postive term followed by two negative terms, and keep dooing this. So the rearranged series begins
1-1/2-1/4+1/3-1/6-1/8...
and notice that we'll never run out of either positive or negative terms since there are infinitely many of both of them. Now put parentheses around the results in this way:
(1-1/2)-1/4+(1/3-1/6)-1/8...
and realize that the numbers inside pairs of parentheses can be simplified:
1/2-1/4+1/6-1/8...
And then further realize that we could factor out 1/2:
(1/2)(1-1/2+1/3-1/4 ....)
which certainly shows that this series converges and its sum is (1/2)S.

Most people find this startling and at least moderately unpleasant. In fact, absolutely convergent series can be rearranged in any way, and rearrangement won't change the sum -- the resulting series will have the same sum as the original series. But any conditionally convergent series can be rearranged so that the sequence of partial sums does almost anything you want. For example, it can be rearranged so that the rearrangement converges to any selected number. (This is because that positive and negative parts of the series separately diverge, but a proof of all this is not really a part of this course.)

Convergence and infinite series can be rather subtle. But now we need to turn to preparing for the final exam.

Using series instead of L'H
I asked people to use Maclaurin series to compute 

     (sin(3x)-3x)2
lim ---------------
x→0   (e2x-1-2x)3
We did this by using the known series for the sine and the exponential functions and then factoring out and canceling as many powers of x as we could (6) from the top and bottom. Solution using L'H would need 6 differentiations of the top and bottom functions, and that's quite unappealing.

Finding some terms of a Taylor series
I asked for the first four non-zero terms of the Maclaurin series for cos(3x)/(1+x4). This can be obtained by using series for cos(3x) (easy to get from the cosine series) and by realizing the 1/(1+x4) is the sum of a geometric series with first term 1 and ratio x4.

An integral
Here's a problem from the textbook: write ∫0xln(1+t2)dt as a power series in x assuming |x|<1. How many terms of this series would be needed to get the value of the integral when x=1/3 to an accuracy of .001? (Of course, I can find an antiderivative of ln(1+t2) using integration by parts fairly easily.)

If you want to come to the review session ...
On Monday afternoon, I'll have a review session from 3 to 5 PM in Hill 525. You could prepare for it by filling out this neat quiz written by Dr. Julia Wolf. She gave it to her class and allowed half an hour. So give yourself half an hour, and then come to the review session with your results.