Zerotoepsilon

 Abel's partial summation formula is a very useful tool in analytic number theory to estimate the behavior of series. It converts the discrete sum into some integral.  Following link contain a nice proof with some example, which will make you comfortable with estimating various sums using this formula. Click here 👇👇👇 Abel Partial Summation Formula

The development of Ring Theory Have you ever thought about the boring definition of rings? Where does it come from? Just to make our life complicated😂....well no Let's make it interesting... Many  books on Abstract Algebra will contain the definition of a ring but  what will be missing are the reasons systems satisfying these particular axioms have been singled out for such intensive study.  What motivated this abstract definition of a ring? Here is an interesting article by  J J O'Connor and E F Robertson,  which concerned  with the development of the theory of commutative rings. Click here to read the article  👇👇👇 The development of Ring Theory

Please go through this article once, it will clear your viewpoint about these two structures. Click here - INTRODUCTORY NOTES ON MODULES

 Hello everyone, as you know that all the colleges/institutes are closed for the last four months due to COVID-19. Students of graduating batches are suffering most as they have to prepare for Ph.D. entrances.  Many of the students nowadays are preparing for GATE/CSIR etc. at their home. So I thought that providing mock tests at no cost will boost their confidence as well as check their preparation. The mock test took a week to complete, I prepared this on google form as it is lesser data consuming and accessible in low internet connection also. Immediately after submitting the exam one can see his score. In the actual exam, there is negative marking and time limit, I did not put any time limit as many people may have poor connectivity. There is no negative marking but after submitting the exam one can see the question paper with the answer key and his response, so he can subtract partial marks for the wrong answer as per gate official guidelines to check his actual score. The...

As we know that all cyclic groups are generated by a single element but what about non-cyclic groups?    How far any non-cyclic group is in the number of generators to cyclic groups? After going through the article (link attached below) you will know some interesting facts as well as the answer to the above questions.  Generating Set for Various Groups Click on the above link.

Open set(so closed set) are building blocks for a Topological space.  A random subset of a topological space (X,  τ)  need not be either open or closed but there is always an open and closed set associated with it. Let A   X where (X,  τ ) is a topological space. We know that  Φ  is an open set i.e. contained in every subset of X. Can we go for an open set larger than  Φ , contained in A? Similiarly, the smallest open set in which contain A. Same thing we can do for the closed set, largest and smallest set respectively contained and containing A. These questions give rise to the concept of Closure and Interior. Closure of A   X  -:  The closure of A in (X,  τ ), is defined as the smallest closed set, which contains A. For example, closure of [0,2)  is [0,2] and closure of set S = {1/n : n    } is S   {0} with usual topology on  .  Some of the text use the following def...

Hello! Here I am going to discuss some relation between Cauchy Sequences and functions. The main motive is to know what kind of function(continuous/uniform) preserves Cauchy sequences and vice versa. ➤ Does a continuous function always preserve a Cauchy sequence? We know sequential def. of continuity, so if we take a Cauchy sequence x_n on a closed subset of   ℝ  then it will surely converge to some point x in the domain, hence f(x_n) will converge to f(x). So our main goal is to deal with subsets that are not closed( they must have a Cauchy sequence that doesn't converge). So one such set is (0,1). Now take f(x) = 1/x and consider x_n = 1/n be a Cauchy sequence. f(x_n) = n, which is not Cauchy hence Continuous function need not preserve Cauchy sequence. ➤  Does a uniformly continuous function always preserve a Cauchy sequence? Suppose f: 𝔻 →  ℝ is uniformly continuous.  By def. of U.C. function, given  ε   > 0 ∃  a  δ ...