1. Paul J. Bleau
Dr. Brendan Sullivan
MATH 2103*01
12/13/15
1. History of the Basel Problem
a. 1644: Pietro Mengoli puts forth the challenge of finding the exact value (not
simply an estimate) of the precise summation of the reciprocals of the squares of
the natural numbers.
b. 1689: The Bernoulli Brothers, Jakob and Johann, face the infinite series. Jakob
particularly was interested in the divergence of infinite series (and of infinite
series in general) and coins the phrase “Basel Problem” in reference to their
hometown, which was also the home town of the problem’s eventual solver.
c. 1734: Leonhard Euler gains immediate fame by solving the Basel Problem at the
age of 28 and presenting it in The Saint Petersburg Academy of Science. Euler’s
proof uses Trigonometry and Factorials to find the exact sum to be
𝜋2
6
.
d. 2015: Paul Bleau aces his 2nd Calc III Project by following Euler’s discovery
using multivariable Calculus
2. 𝐼 = ∬
1
1−𝑥𝑦𝑅
𝑑𝐴 = ∬ 1 + (𝑥𝑦)1
+ (𝑥𝑦)2
+ (𝑥𝑦)3
+ (𝑥𝑦)4
+ ⋯ 𝑑𝐴𝑅
= ∬ 1 + 𝑥𝑦 + 𝑥2
𝑦2
+ 𝑥3
𝑦3
+ 𝑥4
𝑦4
+ ⋯ 𝑑𝐴
𝑅
3. ∫ ∫ 1 + 𝑥𝑦 + 𝑥2
𝑦2
+ 𝑥3
𝑦3
+ 𝑥4
𝑦4
+ ⋯ 𝑑𝑥𝑑𝑦
1
0
1
0
8. = (𝑧 +
𝑧2
8
+
𝑧3
27
+
𝑧4
64
+
𝑧5
125
+ … |
1
0
= ((1 +
1
8
+
1
27
+
1
64
+
1
125
+ ⋯ ) − (0 + 0 + 0 + 0 + 0 + ⋯ ))
= 1 +
1
8
+
1
27
+
1
64
+
1
125
+ ⋯
We know that ∑
1
𝑛3 =
1
13 +
1
23 +
1
33 +
1
43 +
1
53 + ⋯ =∞
𝑛=1 1 +
1
8
+
1
27
+
1
64
+
1
125
+ ⋯
∴ 𝑇 = ∑
1
𝑛3
∞
𝑛=1
Unfortunately, this value is not yet known. Leonhard Euler made an attempt but the
closest he got was determining ∑
(−1) 𝑘
(2𝑘+1)3
∞
𝑛=1 =
𝜋3
32
.
b. However, the value of ∑
1
𝑛 𝑘
∞
𝑛=1 can be determined for all real, even k values, thanks to
Euler and methods used in this project. The value of ∑
1
𝑛 𝑘
∞
𝑛=1 is given by
2(𝑘−2)
𝜋 𝑘
3(2𝑘−2)!
for all
real, even values of k.
c. Not a problem on the Project Page but I noticed that the amount of projects done in our
Calc classes is relative to n – 1 for all Calculus n…Def Eq’s doesn’t have 3, does it?!
9. Bibliography
- Bhand, Ajit. "The Basel Problem and Euler's Triumph." Talk Math. Wordpress, 8
Nov. 2010. Web. 7 Dec. 2015.
- Sangwin, C. J. "An Infinite Series of Surprises." Plus Math. Plus Magazine, 1 Dec.
2001. Web. 7 Dec. 2015.
- Sullivan, Brendan. "The Basel Problem: Numerous Proofs." Carnegie Melon
University, 11 Apr. 2013. Web. 7 Dec. 2015.