devotion & mathematics

Formal power series: In calculus an infinite series of the form 𝜮 a(n)  = a(0) + a(1) x + a(2) x² +...+ a(n) x +... is called a power series in x. Here both x and a(n) are real or complex numbers. To each power series there corresponds a radius of convergence r > 0 such that the series converges absolutely if | x |<r and diverges if | x | > r. Note:          Here the radius r can be +∞ Here in this, we consider power series from a different point of view. We call these power series as FORMAL power series to distinguish them from the ordinary power series of calculus. In the formal power series, x is never assigned a numerical value. In power series 𝜮 c(n) x", the symbol x" is simply a device for locating the position of the nth coefficient a(n). The coefficient a(0) is called the constant coefficient of the series. Let A(x)=a(n) x"; B(x) = b(n) x. Then 1. A(x)+B(x) iff a(n) = b(n) for all n > 0 2. A(x)+B(x)=(a(x)+b(x)) x". 3. A(x) B(x) = c(n) ...

                             INFINITE SERIES                                               Introduction                         P-Test                           Comparision test                          Limit Comparison Test                                                                      Cauchy's nth root test                                    ...

  REAL ANALYSIS- INTRODUCTION :                                                                  Real Analysis is a fundamental branch of mathematics that it rigorously studies real numbers, sequences, series, limits, continuity, differentiation and integration. It provides a formal frame work for understanding calculus, ensuring that intuitive concepts like limits and continuity are precisely defined and logically sound.                                             At its core , real analysis answers fundamental questions such as   * What does it mean for a sequence to converge?  * how do we rigorously define continuity and differentiability?  * What properties make functions well...