Math 4111 Exam 3 review

Relations between series and topology (compactness, closure, etc.)

Limit points $E'=\{x\in\mathbb{R}:\forall r>0, B_r(x)\backslash\{x\}\cap E\neq\phi\}$

Closure $\overline{E}=E\cup E'=\{x\in\mathbb{R}:\forall r>0, B_r(x)\cap E\neq\phi\}$

$p_n\to p\implies \forall \epsilon>0, \exists N$ such that $\forall n\geq N, p_n\in B_\epsilon(p)$

Some interesting results

Lemma

$p\in \overline{E}\iff \exists (p_n)\subseteq E$ such that $p_n\to p$

$p\in E'\iff \exists (p_n)\subseteq E\backslash\{p\}$ such that $p_n\to p$ (you cannot choose $p$ in the sequence)

Bolzano-Weierstrass Theorem

Let $E$ be a compact set and $(p_n)$ be a sequence in $E$. Then $\exists (p_{n_k})\subseteq (p_n)$ such that $p_{n_k}\to p\in E$.

Rudin Proof:

Rudin’s proof uses a fact from Chapter 2.

If $E$ is compact, and $S\subseteq E$ is infinite, then $S$ has a limit point in $E$ ($S'\cap E\neq\phi$).

Examples of Cauchy sequence that does not converge

Cauchy sequence in $(X,d),\forall \epsilon>0, \exists N$ such that $\forall m,n\geq N, d(p_m,p_n)<\epsilon$

Let $X=\mathbb{Q}$ and $(p_q)=\{1,1.4,1.41,1.414,1.4142,1.41421,\dots\}$ The sequence is Cauchy but does not converge in $\mathbb{Q}$.

This does not hold in $\mathbb{R}$ because compact metric spaces are complete.

Fact: Every Cauchy sequence is bounded.

Proof that $e$ is irrational

$e=\sum_{n=0}^\infty \frac{1}{n!}$

Let $s_n=\sum_{k=0}^n \frac{1}{k!}$

So $e-s_n=\left(\sum_{k=n+1}^\infty \frac{1}{k!}\right)<\frac{1}{n!n}$

If $e$ is rational, then $\exists p,q\in\mathbb{Z}$ such that $e=\frac{q}{p}$ and $q!s_q\in\mathbb{Z}$, $q!e=q!\frac{p}{q}\in \mathbb{Z}$, so $q!(e-s_q)\in\mathbb{Z}$

$0<q!(e-s_q)<\frac{1}{n!n}$ leads to contradiction.

$\limsup$ and $\liminf$

Let $(a_n)=(-1)^n$

$\limsup a_n=1$ and $\liminf a_n=-1$

Let $(a_n)\to a$

$\limsup a_n=\liminf a_n=a$

Facts about $\limsup$ and $\liminf$

Convergence of subsequence

$\limsup$ is the largest value that subsequence of $a_n$ can approach to.

$\liminf$ is the smallest value that subsequence of $a_n$ can approach to.

Elements of sequence

$\forall x>s^*,\{n:a_n>x\}$ is finite. $\exists N$ such that $\forall n\geq N, a_n\leq x$

$\forall x<s^*,\{n:a_n>x\}$ is infinite.

One example is $(a_n)=(-1)^n\frac{n}{n+1}$

$\limsup a_n=1$ and $\liminf a_n=-1$

So the size of set of elements of $a_n$ that are greater than any $x<1$ is infinite. and the size of set of elements of $a_n$ that are greater than any $x>1$ is finite.

$\limsup(a_n+b_n)\leq \limsup a_n+\limsup b_n$

One example for smaller than is $(a_n)=(-1)^n$ and $(b_n)=(-1)^{n+1}$

$\limsup(a_n+b_n)=0$ and $\limsup a_n+\limsup b_n=2$

($\forall n,s_n\leq t_n$) $\implies \limsup s_n\leq \limsup t_n$

One example of using this theorem is $(s_n)=\left(\sum_{k=1}^n\frac{1}{k!}\right)$ and $(t_n)=\left(\frac{1}{n}+1\right)^n$

Rearrangement of series

Will not be tested.

infinite sum is not similar to finite sum. For infinite sum, the order of terms matters. But for finite sum, the order of terms does not matter, you can rearrange the terms as you want.

Ways to prove convergence of series

n-th term test (divergence test)

If $\lim_{n\to\infty}a_n\neq 0$, then $\sum a_n$ diverges.

Definition of convergence of series (convergence and divergence test)

If $\sum a_n$ converges, then $\lim_{n\to\infty}\sum_{k=1}^n a_k=0$.

Example: Telescoping series and geometric series.

Comparison test (convergence and divergence test (absolute convergence))

Let $(a_n)$ be a sequence in $\mathbb{C}$ and $(c_n)$ be a non-negative sequence in $\mathbb{R}$. Suppose $\forall n, |a_n|\leq c_n$.

(a) If the series $\sum_{n=1}^{\infty}c_n$ converges, then the series $\sum_{n=1}^{\infty}a_n$ converges.
(b) If the series $\sum_{n=1}^{\infty}a_n$ diverges, then the series $\sum_{n=1}^{\infty}c_n$ diverges.

Ratio test (convergence and divergence test (absolute convergence))

$\left|\frac{a_{n+1}}{a_n}\right| \leq \alpha \implies |a_n|\leq \alpha^n$

Given a series $\sum_{n=0}^{\infty} a_n$, $a_n\in\mathbb{C}\backslash\{0\}$.

Then

(a) If $\limsup_{n\to\infty} \left|\frac{a_{n+1}}{a_n}\right| < 1$, then $\sum_{n=0}^{\infty} a_n$ converges.
(b) If $\left|\frac{a_{n+1}}{a_n}\right| \geq 1$ for all $n\geq n_0$ for some $n_0\in\mathbb{N}$, then $\sum_{n=0}^{\infty} a_n$ diverges.

Root test (convergence and divergence test (absolute convergence))

$\sqrt[n]{|a_n|} \leq \alpha \implies |a_n|\leq \alpha^n$

Given a series $\sum_{n=0}^{\infty} a_n$, put $\alpha = \limsup_{n\to\infty} \sqrt[n]{|a_n|}$.

Then

(a) If $\alpha < 1$, then $\sum_{n=0}^{\infty} a_n$ converges.
(b) If $\alpha > 1$, then $\sum_{n=0}^{\infty} a_n$ diverges.
(c) If $\alpha = 1$, the test gives no information

Cauchy criterion

Geometric series

P-series

(a) $\sum_{n=0}^{\infty}\frac{1}{n}$ diverges.
(b) $\sum_{n=0}^{\infty}\frac{1}{n^2}$ converges.

Cauchy condensation test (convergence test)

Suppose $(a_n)$ is a non-negative sequence. The series $\sum_{n=1}^{\infty}a_n$ converges if and only if the series $\sum_{k=0}^{\infty}2^ka_{2^k}$ converges.

Dirichlet test (convergence test)

Suppose

(a) the partial sum $A_n$ of $\sum a_n$ form a bounded sequence.
(b) $b_0\geq b_1\geq b_2\geq \cdots$ (non-increasing)
(c) $\lim_{n\to\infty}b_n=0$.

Then $\sum a_nb_n$ converges.

Example: $\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n}$ converges.

Abel’s test (convergence test)

Let $(b_n)^\infty_{n=0}$ be a sequence such that:

(a) $b_0\geq b_1\geq b_2\geq \cdots$ (non-increasing) (b) $\lim_{n\to\infty}b_n=0$

Then if $|z|=1$ and $z\neq 1$, $\sum_{n=0}^\infty b_nz^n$ converges.