Theorems

Real Sequences

Sequence of real numbers A sequence of real numbers (or a real sequence) is defined as a function $ f: \mathbb{N} \to \mathbb{R}$ , where $ \mathbb{N}$ is the set of natural numbers and $ \mathbb{R}$ is the set of real numbers. Thus, $ f(n)=r_n, \ n \in \mathbb{N}, \

Fermat Numbers

Fermat Numbers, a class of numbers, are the integers of the form $ F_n=2^{2^n} +1 \ \ n \ge 0$ . For example: Putting $ n := 0,1,2 \ldots$ in $ F_n=2^{2^n}$ we get $ F_0=3$ , $ F_1=5$ , $ F_2=17$ , $ F_3=257$ etc. Fermat observed that all

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} $ ,

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