Homework #3

EE 503: Summer 2026

ImportantAssignment Details

Assigned: 08 June
Due: Monday, 15 June at 12:00

BrightSpace Assignment: Homework 3

Daily Derivation

Prove the Binomial Theorem, the Borel-Cantelli Lemmas, and Infinite Boole’s inequality.

TipInstructions

Write your solutions to these homework problems. Submit Handout and Book separately on BrightSpace by the due date. Show all work and box answers where appropriate. Do not guess.

Handout

Problem 1

A function \(f: X \rightarrow Y\) is onto (or surjective) iff for each “image” element \(y \in Y\) there is a “pre-image” element \(x \in X\) such that \(y = f(x)\). A function \(f\) is one-to-one (or injective) iff distinct pre-images have distinct images: \(f(x_1) \ne f(x_2)\) if \(x_1 \ne x_2\) for all \(x_1 \in X\) and all \(x_2 \in X\). Note that the contrapositive of the last statement states that \(f(x_1) = f(x_2)\) only if \(x_1 = x_2\) for all \(x_1 \in X\) and all \(x_2 \in X\). A function is bijective iff it is both injective and surjective (precisely when the inverse point function \(f^{-1}\) exists). Suppose \(A \subset X\) and \(B \subset Y\) for \(f: X \rightarrow Y\). Then prove or disprove:

  1. \(f(f^{-1}\{B\}) \subset B\).

  2. \(f(f^{-1}\{B\}) = B\) if \(f\) is surjective.

  3. \(A \subset f^{-1}\{f(A)\}\).

  4. \(A = f^{-1}\{f(A)\}\) if \(f\) is injective.

  5. \(f: X \rightarrow Y\) is bijective implies \(f:2^X \rightarrow 2^Y\) is bijective.

Problem 2

Use the ratio test to determine whether the following infinite series diverge or converge:

  1. \(\displaystyle \sum_{n=1}^{\infty} \frac{(n+1)!}{n^n}\).

  2. \(\displaystyle \sum_{n=1}^{\infty} \frac{1 \cdot 3 \cdot 5 \cdots (2n - 1)}{n!}\).

  3. \(\displaystyle \sum_{n=1}^{\infty} (-1)^n \frac{3^{2n+1}}{(2n+1)!}\).

  4. \(\displaystyle \sum_{n=1}^{\infty} \frac{3^n \cdot n!}{n^n}\).

Problem 3

Find the interval of convergence for these power series (check both endpoints):

  1. \(\displaystyle \sum_{n=1}^{\infty} (-1)^n \frac{(x - e)^n}{n \cdot e^n}\).

  2. \(\displaystyle \sum_{n=1}^{\infty} \frac{(x - 1)^{2n}}{n \cdot 4^n}\).

  3. \(\displaystyle \sum_{n=2}^{\infty} \frac{x^n}{n \ln n}\).

Problem 4

Use the \(\epsilon\)-definition (i.e. garden hose) to evaluate the limit of these sequences. Given \(\epsilon = 10^{-6}\) what is the smallest index \(n_0\) such that \(|a_n - L| < \epsilon\) for all \(n \ge n_0\)?

  1. \(\displaystyle \frac{n^2 - 5n + 6}{2n^2 + 7}\).

  2. \(\displaystyle \sqrt[3]{n^3 + n^2} - n\).

Problem 5

Let \(A \times B = \{(x,y): x \in A \text{ and } y \in B\}\). Suppose \(A = \{a_1, a_2, a_3\}\) and \(B = \{b_1, b_2\}\). Then what is the Cartesian product \(A \times B\)? How many elements in \(2^{A \times B}\)? Produce four sub-collections \(\mathcal{A} \subset 2^{A \times B}\) that are sigma-algebras.

Problem 6

Prove or disprove:

  1. If \(A \subset X\) and \(B \subset Y\) then \(A \times B \subset X \times Y\).

  2. \((A \cup B) \times C = (A \times C) \cup (B \times C)\).

Book

Leon-Garcia

Chapter 2: 44-47, 58-61