MTH631 Assignment 1 Solution Fall 2026

This assignment covers fundamental concepts in real analysis, specifically focusing on the uniform convergence of series and sequences of functions. Students are required to apply various tests, including Cauchy's uniform convergence criterion, Dirichlet's test, and the Weierstrass M-test, to analyze the convergence properties of given series. Additionally, the assignment explores the interchange of limits with integration and differentiation for sequences of functions. Students must provide detailed mathematical proofs and calculations for each problem.

1. 1

   Question 1

   Apply Cauchy's uniform convergence criterion to the series of functions defined by f_n(x) = 1/(n+x^2) - 1/(n+1+x^2). Show that the series converges uniformly on the interval (-∞, ∞) and determine the smallest integer N such that the norm of the partial sum is less than 0.01 for all m ≥ n ≥ N.

2. 2

   Question 2

   Use Dirichlet's test to demonstrate that the series of cos(nx)/n converges uniformly on the interval S = [π/3, 5π/3]. You must also calculate a numerical bound M for the partial sums of cos(kx) on the specified interval S.

3. 3

   Question 3

   For the series of functions (x/(2+x))^n, identify all values of x for which the series converges pointwise and determine its sum F(x). Then, apply the Weierstrass M-test to prove uniform convergence on the interval [-1/2, 5], explicitly stating the sequence M_n and the value of the sum of M_n.

4. 4

   Question 4

   Given F_n(x) = x/(1 + nx^2) on [0, 1], verify the interchange of limit and integration by computing and comparing the limit of the integral and the integral of the limit. Furthermore, compute F_n'(0) and F'(0) to determine if the interchange of limit and differentiation holds, justifying your conclusion by identifying which hypothesis fails.

What to submit

One Word file using Math Type or Equation Editor for all mathematical symbols and equations.

Topics covered

Uniform convergence Cauchy's criterion Dirichlet's test Weierstrass M-test Pointwise convergence Interchange of limit and integration Interchange of limit and differentiation