devotion & mathematics

 Historical Introduction to Number Theory : Part -2                 ( part 1 continuation )                                                          The following are few examples of Pythagorean triples (3,4,5), (5,12,13), (7,24,25), (9,40,41), (11,60,61), (13, 84,85), (15,112,113),(17,144,145) (19,180,181) etc. There are other Pythagorean triples besides these ; for example : (8,15,17), (12,35,3),(16,63,65) (20,99,101) etc. In these examples we have z=y+2. Plato found a method for determining all these triples ; in modern notation they are given by the formulae X=4n,   y= 4n 2 -1,   z= 4n 2 +1. Around 300 BC an important event occurred in the history of mathematics. The appearance of Euclid’s Elements, a collection of 13 books , transformed mathematics from numerology into a ...

  Historical Introduction  to Number Theory :  Part 1                                                                                                             The theory of numbers is that branch of mathematics which deals with properties of the whole numbers 1,2,3,...  also called the counting numbers or positive integers.                 The positive integers are undoubteldly man's first mathematical creation. It is hardly possible to imagine human beings without the ability to count, at least within a limited range. Historical record shows that as early as 5700 BC the ancient sumerians kept a calender, so they must have developed some  form of ari...

                              Limit Comparison Test:                             Proof                       problems & Solutions                                                                                                        Back  

  Limit comparison test :                                                                                                                                                                              Back Statement : If Σu n and Σv n are two series of non-negative terms such that  lim (u n /v n ) = l. Then a)         If l ≠ 0, then Σu n and Σv n converge or diverge together and b)       If l = 0 and if   Σv n convergent then Σu n convergent . P     Proof #limit #compari...

Problems & Solutions :   Test for convergence of Σ (2n-1)/n(n+1)(n+2).                                                                Back Solution : Test for convergence of  1+ 4/5 + 6/10+ 8/17+ …+ 2n/(n 2 +1)+ … Solution: Test for convergence of  1.2/3.4.5 + 2.3/4.5.6+ 3.4/5.6.7 + … Solution : Test for convergence of Σ1/ n^( 𝜶+( 𝜷/n)). Solution : Test for convergence of Σ 1/( 2 n +3 n ). Solution : Test for convergence of Σ (√(n+1) - √n ). Solution : Test for convergence of Σ ( √ (n 2 +1) – n ). Solution : Test for convergence of  Σ ( √(n 3 +1) - √( n 3 )  ). Solution :  Test for convergence of Σ ( √ (n 4 +1) - √ ( n 4 -1) ).

 Cauchy’s n th root test or Root test :                                                                                                                                                                               Back   Statement :                               If Σ u n  is a series of positive terms such that lim u n 1/n =l, then (a)     Σ u n converges if l<1 and (b) Σ u n diverges if l>1. Proof :  

 Cauchy's nth root test problems & solutions :                                                                                                                                         Back 1.        Test for convergence of (1/3)+ (2/5) 2 + (3/7) 3 +…        Solution : 1.        Test for convergence of Σ 2 n /n 3 .        Solution : 1.        Test for convergence of Σ x n /n n.          Solution : 1.        Test for convergence of Σ (1+ 1/n) -n .     ...