# Real Analysis

## Dedekind’s Theory of Real Numbers

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…

## 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…

## Derivative of x squared is 2x or x ? Where is the fallacy?

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…