# Real Analysis

## Numbers – The Basic Introduction

If mathematics was a language, logic was the grammar, numbers should have been the alphabet. There are many types of numbers we use in mathematics, but at a broader aspect we may categorize them in two categories: 1. Countable Numbers 2. Uncountable Numbers The numbers which can be counted in nature are called Countable Numbers

## Just another way to Multiply

Multiplication is probably the most important elementary operation in mathematics; even more important than usual addition. Every math-guy has its own style of multiplying numbers. But have you ever tried multiplicating by this way? Exercise: $88 \times 45$ =? Ans: as usual :- 3960 but I got this using a particular way: 88

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 $f(x)= \displaystyle{\sum_{n=0}^{\infty} } b^n \cos (a^n ## Free Online Calculus Text Books In this list I have collected all useful and important free online calculus textbooks mostly in downloadable pdf format. Feel free to download and use these. Elementary Calculus : An approach using infinitesimals by H. J. Keisler https://www.math.wisc.edu/~keisler/keislercalc-12-23-18.pdf Multivariable Calculus by Jim Herod and George Cain http://people.math.gatech.edu/~cain/notes/calculus.html Calculus by Gilbert Strang http://ocw.mit.edu/ans7870/textbooks/Strang/strangtext.htm Calculus Bible by ## Dedekind’s Theory of Real Numbers Intro Let$ \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}$. Thus, the ## D’Alembert’s Ratio Test of Convergence of Series In this article we will formulate the D’ Alembert’s Ratio Test on convergence of a series. Let’s start. Statement of D’Alembert Ratio Test A series$ \sum {u_n}$of positive terms is convergent if from and after some fixed term$ \dfrac {u_{n+1}} {u_n} < r < {1} $, where r is a fixed ## Dirichlet’s Theorem and Liouville’s Extension of Dirichlet’s Theorem Topic Beta & Gamma functions Statement of Dirichlet’s Theorem$ \int  \int  \int_{V}  x^{l-1} y^{m-1} z^{n-1} dx  dy ,dz = \frac { \Gamma {(l)} \Gamma {(m)} \Gamma {(n)} }{ \Gamma{(l+m+n+1)} } $, where V is the region given by$ x \ge 0 y \ge 0 z \ge 0  x+y+z \le 1 $. ## Derivative of x squared is 2x or x ? Where is the fallacy? We all know that the derivative of$x^2\$ is 2x. But what if someone proves it to be just x?

We are now on Telegram!