Weierstrass had drawn attention to the fact that there exist functions which are continuous for every value of $ x$ but do not possess a derivative for any value. We now consider the celebrated function given by Weierstrass to show this fact. It will be shown that if $ f(x)= \displaystyle{\sum_{n=0}^{\infty} } b^n \cos (a^n…

# Math

**Mathematics Blog by Gaurav Tiwari.** This section is dedicated to exploring various mathematical concepts and theories, transcending the boundaries of traditional high school and early college curriculums.

The mathematics topics here encompass various subfields, including algebra, calculus, number theory, geometry, topology, and more.

In this list, I have collected all useful and important free online calculus textbooks, mostly in downloadable pdf format. Feel free to download and use these. Elementary Calculus : An approach using infinitesimals by H. J. Keisler https://www.math.wisc.edu/~keisler/keislercalc-12-23-18.pdf Multivariable Calculus by Jim Herod and George Cain http://people.math.gatech.edu/~cain/notes/calculus.html Calculus by Gilbert Strang http://ocw.mit.edu/ans7870/textbooks/Strang/strangtext.htm Calculus Bible by…

Let $ \mathbf{Q}$ be the set of rational numbers. It is well known that $ \mathbf{Q}$ is an ordered field and also the set $ \mathbf{Q}$ is equipped with a relation called “less than” which is an order relation. Between two rational numbers there exists an infinite number of elements of $ \mathbf{Q}$. Thus, the system…

Looking for free PDF algebra and topology books online? I have gathered a list of free e-books on Algebra, Topology, and Related Mathematics. I hope it will be very helpful to all students and teachers searching for high-quality content. If any link is broken, please email me at gaurav@gauravtiwari.org. Let’s start. Abstract Algebra Online by Prof. Beachy This…

In this article, we will formulate the D’ Alembert’s Ratio Test on convergence of a series. Let’s start. Statement of D’Alembert Ratio Test A series $ \sum {u_n}$ of positive terms is convergent if from and after some fixed term $ \dfrac {u_{n+1}} {u_n} < r < {1} $ , where r is a fixed…

Problem solving is more than just finding answers. Learning how to solve problems in mathematics is simply to know what to look for. Mathematics problems often require established procedures. To become a problem solver, one must know What, When and How to apply them. To identify procedures, you have to be familiar with the different…

I was reading a book on ancient mathematics problems from Indian mathematicians. Here I wish to share one problem from Bhaskaracharya‘s famous creation Lilavati. Who was Bhaskaracharya? Bhaskara II, who is popularly known as Bhaskaracharya, was an Indian mathematician and astronomer from the 12th century. He’s especially known for the discovery of the fundamentals of…

The Collatz Conjecture is one of the Unsolved problems in mathematics, especially in Number Theory. The Collatz Conjecture is also termed as 3n+1 conjecture, Ulam Conjecture, Kakutani’s Problem, Thwaites Conjecture, Hasse’s Algorithm, and Syracuse Problem. Collatz Conjecture Statement If you keep repeating this procedure, you shall reach the number 1 at last. Illustrations » Starting…

Topic: Beta & Gamma functions The Gamma function and Beta functions belong to the category of special transcendental functions and are defined in terms of improper definite integrals. Definitions of Beta and Gamma functions are given below. But before that, let’s quote the statement of Dirichlet’s theorem so that we can work around Liouville’s extension…

Puzzle Two women were selling marbles in the market place — one at three for a Rupee and other at two for a Rupee. One day both of then were obliged to return home when each had thirty marbles unsold. They put together the two lots of marbles and handing them over to a friend…

In 1904, the french Mathematician Henri Poincaré (en-US: Henri Poincare) posed an epoch-making question, which later came to be termed as Poincare Conjecture, in one of his papers, which asked: If a three-dimensional shape is simply connected, is it homeomorphic to the three-dimensional sphere? Henri Poincare – 1904 So what does it really mean? How…

Derivative of x squared As we know, the derivative of x squared, i.e., differentiation of $ x^2$ , with respect to $ x$, is $ 2x$. i.e., $ \dfrac{d}{dx} x^2 = 2x$ A Curious Case Suppose we write $ x^2$ as the sum of $ x$ ‘s written up $ x$ times. i.e., $ x^2…