Real Analysis Problems [Sample Final Questions]

In this page, we list Real Analysis problems that can be treated as a sample question for the upcoming final exams.

Real Analysis Practice Problems

Notation:

• $\mathbb{N}$:= The set of natural numbers
• $\mathbb{Z}$:= The set of integers

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 the below sets:

1. $\mathbb{Z}$
2. $\mathbb{N} \cup \mathbb{Z}$.
3. The open interval (0, 1)
4. The set of rational numbers.

Q4: Find the limit points of the sets

1. $\left\{\dfrac{1}{n}: n \in \mathbb{N} \right\}$
2. $\left\{1+\dfrac{1}{n}: n \in \mathbb{N} \right\}$

Q5: Find the Sup and Inf of the sets

1. $\left\{\dfrac{1}{n}: n \in \mathbb{N} \right\}$
2. $\left\{\dfrac{1}{m}+\dfrac{1}{n}: m, n \in \mathbb{N} \right\}$
3. $\left\{\dfrac{1}{2^m}+\dfrac{1}{3^n}: m, n \in \mathbb{N} \right\}$

Q6: Prove that lim_n→∞ n^1/n = 1.

Q7: Find the value of the series $1+\dfrac{1}{1\cdot 2}+\dfrac{1}{2}+\dfrac{1}{2\cdot 3}$ $+\dfrac{1}{2^2}+\dfrac{1}{3\cdot 4}+\dfrac{1}{2^3}+\cdots$

Q8. State limit comparison test, ratio test and root test for an infinite series.

Q9. Prove that the series

1. $\displaystyle \sum_{n=1}^\infty \dfrac{n}{2n+1}$ is not convergent.
2. $\displaystyle \sum_{n=1}^\infty \dfrac{n}{2n^2+n}$ is convergent.

Q10. Test the convergence of the following series:

1. $\sum \sin \left( \dfrac{1}{n} \right)$ [Full Solution]
2. $\sum \dfrac{1}{n} \sin \left( \dfrac{1}{n} \right)$ [Full Solution]
3. $\sum_{n=1}^\infty \dfrac{n}{n^2+\sqrt{n}}$.
4. $\sum_{n=1}^\infty \dfrac{n^{n^2}}{(n-1)^{n^2}}$. [Full Solution]

Q11: State Leibnitz’s test. Test the convergence of the series below:

(i) $\dfrac{1}{2}-\dfrac{1}{4}+\dfrac{1}{6}-$ $\dfrac{1}{8}+\cdots$

(ii) $1-\dfrac{1}{2^2}+\dfrac{1}{3^2}-$ $\dfrac{1}{4^2}+\cdots$

Q12: Show the series $\displaystyle \sum_{n=1}^\infty \dfrac{1}{n}$ is not convergent using Cauchy’s principle.