Real Analysis

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?

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…