Math by Tathagata
Uniform Continuity is a special kind of continuity. In this page, we will learn about uniformly continuous functions with examples and related theorems. Notation: I= An interval.ℝ= The set of all real numbers. MOTIVATION: Let f: I → ℝ be a continuous function and c ∈ I. As f is continuous at c, for every … Read more
Continuous functions is an important topic in Real Analysis. On this page, we will study about continuous functions along with its several properties and examples. Definition Definition of a Continuous Function: Let f: D → ℝ be a function and let c ∈ D. We say f is continuous at x=c, if limx→c f(x) = … Read more
The sequences (-1)n and (-1)n/n are special type of sequences. In this page, we will study the convergence of the sequences (-1)^n and (-1)^n/n. Convergence of (-1)n Question: Discuss the convergence of the sequence {(-1)n}. Answer: The sequence {(-1)n} can be written as follows: {(-1)n} = {-1, 1, -1, 1, -1, 1, …}. Therefore, the … Read more
In this page, we list Real Analysis practice problems that can be treated as a sample question for the upcoming final exams. Real Analysis Practice Problems Notation: Q1: Define a rational number. Show $\sqrt{3}$ is an irrational number. Q2: Prove that $\mathbb{N}$ has no limit point. Q3: Define an enumerable set. Discuss the enumerability of … Read more
A series of the form a0+a1x+a2x2+… = ∑ anxn is called a power series. The radius of convergence of a power series is a very powerful tool in many areas of Mathematics. In this article, we will study the convergence of a power series. Definition of Power Series A series of the form a0+a1(x-a)+a2(x-a)2+… = … Read more
A series ∑an is called absolutely convergent if ∑|an| converges. A series ∑an is called conditionally convergent if it converges but not absolutely. In this article, we study absolute and conditional convergence with their definitions and examples. Absolutely Convergent Series Definition: A series $\displaystyle \sum_{n=1}^\infty a_n$ is called absolutely convergent if the series $\displaystyle \sum_{n=1}^\infty … Read more
An alternating series is a series containing terms alternatively positive and negative. One can check its convergence using Leibnitz’s test. In this article, we will study alternating series, Leibnitz’s test with solved problems. Definition of Alternating Series A series of the form $\displaystyle \sum_{n=1}^\infty$ (-1)n+1an, where an>0 for all n, is called an alternating series. … Read more
Cauchy’s Root test for series is used to test the convergence or divergence of an infinite series. It states that if limn→∞ an1/n is less than 1 then the series ∑an converges and if the limit is greater than 1 then the series diverges. In this article, we will study Cauchy root test of a … Read more
The ratio test for series is useful to check a series is convergent or divergent. Here we consider the limit of the quotient of (n+1)-th and n-th terms of the series, and depending on this limit is greater or less than 1 we decide its convergence. In this article, we study the ratio test of … Read more
The comparison test for series is used to check whether the series is convergent or not. Comparison test is useful when one can compare the given series with another series whose convergence is known. In this article, we learn about comparison test of a series with its limit form and some solved examples. Comparison Test … Read more