The document discusses the history and solution of the Basel problem. It begins by outlining the problem's origins in 1644 and its consideration by Bernoulli brothers in 1689. It then notes that Leonhard Euler solved the problem in 1734 by proving the exact sum is π2/6 using trigonometry and factorials. Finally, it states that Paul Bleau successfully recreated Euler's solution using multivariable calculus for a class project in 2015.
Numerical solution of a system of linear equations by
1) LU FACTORIZATION METHOD.
2) GAUSS ELIMINATION METHOD.
3) MATRIX INVERSION BY GAUSS ELIMINATION METHOD.
Numerical solution of a system of linear equations by
1) LU FACTORIZATION METHOD.
2) GAUSS ELIMINATION METHOD.
3) MATRIX INVERSION BY GAUSS ELIMINATION METHOD.
The aim of this paper is to study the existence and approximation of periodic solutions for non-linear systems of integral equations, by using the numerical-analytic method which were introduced by Samoilenko[ 10, 11]. The study of such nonlinear integral equations is more general and leads us to improve and extend the results of Butris [2].
ABSTRACT: In this paper, we construct new classes of derivative-free of tenth-order iterative methods for solving nonlinear equations. The new methods of tenth-order convergence derived by combining of theSteffensen's method, the Kung and Traub’s of optimal fourth-order and the Al-Subaihi's method. Several examples to compare of other existing methods and the results of new iterative methods are given the encouraging results and have definite practical utility.
The aim of this paper is to study the existence and approximation of periodic solutions for non-linear systems of integral equations, by using the numerical-analytic method which were introduced by Samoilenko[ 10, 11]. The study of such nonlinear integral equations is more general and leads us to improve and extend the results of Butris [2].
ABSTRACT: In this paper, we construct new classes of derivative-free of tenth-order iterative methods for solving nonlinear equations. The new methods of tenth-order convergence derived by combining of theSteffensen's method, the Kung and Traub’s of optimal fourth-order and the Al-Subaihi's method. Several examples to compare of other existing methods and the results of new iterative methods are given the encouraging results and have definite practical utility.
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ملزمة الرياضيات للصف السادس العلمي الاحيائي - التطبيقيanasKhalaf4
ملزمة الرياضيات للصف السادس العلمي
الاحيائي التطبيقي
باللغة الانكليزية
لمدارس المتميزين والمدارس الأهلية
Chapter one: complex numbers
Dr. Anas dheyab Khalaf
ملزمة الرياضيات للصف السادس التطبيقي الفصل الاول الاعداد المركبة 2022anasKhalaf4
طبعة جديدة ومنقحة
حل تمارين الكتاب
شرح المواضيع الرياضية بالتفصيل وبأسلوب واضح ومفهوم لجميع المستويات
حلول الاسألة الوزارية
اعداد الدكتور أنس ذياب خلف
email: anasdhyiab@gmail.com
Partial differentiation, total differentiation, Jacobian, Taylor's expansion, stationary points,maxima & minima (Extreme values),constraint maxima & minima ( Lagrangian multiplier), differentiation of implicit functions.
ملزمة الرياضيات للصف السادس الاحيائي الفصل الاولanasKhalaf4
طبعة جديدة ومنقحة
حل تمارين الكتاب
شرح المواضيع الرياضية بالتفصيل وبأسلوب واضح ومفهوم لجميع المستويات
حلول الاسألة الوزارية
اعداد الدكتور أنس ذياب خلف
email: anasdhyiab@gmail.com
Basic concepts of integration, definite and indefinite integrals,properties of definite integral, problem based on properties,method of integration, substitution, partial fraction, rational , irrational function integration, integration by parts, reduction formula, improper integral, convergent and divergent of integration
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.