# Calculus

## 20 Best Calculus Books in 2022 [Best Sellers Included]

Fresh into Calculus? Are you looking for the best calculus books that you should start with? Well, when I was your age I had tried several books from library, borrowed books from friends and purchased over a dozen from Amazon. But not all books were effective and most of these only wasted my time. Going…

## Irrational Numbers and The Proofs of their Irrationality

“Irrational numbers are those real numbers which are not rational numbers!” Def.1: Rational Number A rational number is a real number which can be expressed in the form of where $a$ and $b$ are both integers relatively prime to each other and $b$ being non-zero. Following two statements are equivalent to the…

## The Area of a Disk

If you are aware of elementary facts of geometry, then you might know that the area of a disk with radius $R$ is $\pi R^2$ . The radius is actually the measure(length) of a line joining the center of disk and any point on the circumference of the disk or any other circular…

## Triangle Inequality

Triangle inequality has its name on a geometrical fact that the length of one side of a triangle can never be greater than the sum of the lengths of other two sides of the triangle. If $a$ , $b$ and $c$ be the three sides of a triangle, then neither $a$…

## 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…

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 $f(x)= \displaystyle{\sum_{n=0}^{\infty} } b^n \cos (a^n… ## Free Online Calculus Text Books In this list I have collected all useful and important free online calculus textbooks mostly in downloadable pdf format. Feel free to download and use these. Elementary Calculus : An approach using infinitesimals by H. J. Keisler https://www.math.wisc.edu/~keisler/keislercalc-12-23-18.pdf Multivariable Calculus by Jim Herod and George Cain http://people.math.gatech.edu/~cain/notes/calculus.html Calculus by Gilbert Strang http://ocw.mit.edu/ans7870/textbooks/Strang/strangtext.htm Calculus Bible by… ## D’Alembert’s Ratio Test of Convergence of Series In this article we will formulate the D’ Alembert’s Ratio Test on convergence of a series. Let’s start. Statement of D’Alembert Ratio Test A series$ \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$is 2x. But what if someone proves it to be just x? ## Solving Ramanujan’s Puzzling Problem Consider a sequence of functions as follows:-$ f_1 (x) = \sqrt {1+\sqrt {x} }  f_2 (x) = \sqrt{1+ \sqrt {1+2 \sqrt {x} } }  f_3 (x) = \sqrt {1+ \sqrt {1+2 \sqrt {1+3 \sqrt {x} } } } $……and so on to$ f_n (x) = \sqrt {1+\sqrt{1+2 \sqrt… 