Math 105A Homework 2

Please complete the following from the textbook:

  • Section 2.1 - #1-2
  • Section 2.2 - #1-8
  • Section 2.3 - #1-4

Transcribed Questions

2.1.1

Find what the following sequences converge to and prove they indeed converge to your claimed value.

  • ${x_n} = { 3+\frac{(-1)^n}{2n} }$
  • ${x_n} = { \frac{2n-5}{5n+7} }$
  • ${x_n} = { \frac{2n^2+3n}{n+n^2} }$
  • ${x_n} = { \frac1{n^2} + \frac2{n^2} + \dots + \frac{n}{n^2} }$

2.1.2

Let $h > 0$ be fixed. Prove (either by induction or using the binomial theorem) that \((1+h)^n \geq 1 + nh, \quad n = 1,2,\dots\) Deduce that, if $0 < r < 1$, then $\lim_{n \rightarrow \infty}r^n = 0$. [Hint: explain why you can write $r = \frac1{1+h}$ for some $h > 0$.]

2.2.1

Prove or disprove with counterexample the following variation of Theorem 11: Given two sequences ${x_n}$ and ${y_n}$ that are convergent with $x_n \rightarrow x$ and $y_n \rightarrow y$, if $x_n > y_n \forall n \in \mathbb{N}$, then $x > y$.

2.2.2

Prove both parts of Lemma 12:

  • Suppose for two sequences ${a_n}$ and ${b_n}$ that $0 \leq b_n \leq a_n$ is true for all $n \in \mathbb N$ If $a_n \rightarrow 0$, $b_n \rightarrow 0$
  • For sequences ${a_n}, {b_n}$ and ${c_n}$ with the property that $a_n \leq b_n \leq c_n \ \forall n \in \mathbb N$, if ${a_n}$ and ${c_n}$ are both convergent with $\lim_{n \rightarrow \infty} a_n = L = \lim_{n \rightarrow \infty} c_n$, then ${b_n}$ is convergent with $\lim_{n \rightarrow \infty} b_n = L$

2.2.3

Conjecture the value of the following and provide a proof of its convergence (if it does?) \(\lim_{n \rightarrow \infty} \sqrt{n^2 + 6n} - n\) Hint: Finding a lower bound on a term in a denominator give an upper bound on its reciprocal, i.e. $f(x) > M$ means $\frac1{f(x)} < \frac1M$

2.2.4

  • Suppose that the sequence ${x_n}$ is bounded and that $y_n \rightarrow 0$. Prove that ${x_ny_n}$ converges to $0$.
  • Give an example in which $y_n \rightarrow 0$, and $x_ny_n$ does not converge to 0.

2.2.5

  • Suppose that $x_n \rightarrow x$, and define $y_n$ to be the sequence given by
\[y_n = \frac{x_1 + x_2 + \dots + x_n}{n} = \frac1n \sum_{k=1}^n x_k\]

i.e. the arithmetic mean of the first $n$ terms. Show that $y_n \rightarrow x$

  • Let ${x_n} = {(-1)^n}$. We know that $x_n$ diverges, but show that $y_n$ in this instance converges.

2.2.6

  • Prove that if ${x_n}$ converges to $L$, then ${|x_n|}$ converges to $|L|$
  • Is there converse always true? (is it always the case that if ${|x_n|}$ converges then ${x_n}$ converges?)

2.2.7

A polynomial with rational coefficients $p(x)$ is given by \(p(x) = a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x + a_0, \ \ a_0,a_1,a_2,\dots,a_n \in \mathbb Q\) and a rational function $R(x)$ is a fraction of two polynomials with rational coefficients, i.e. $R(x) = \frac{p(x)}{q(x)}$

  • Using induction on the degree of the polynomial and the algebraic limit laws proven in this section, prove why if $x_n \rightarrow x$, then $p(x_n) \rightarrow p(x)$ for any polynomial $p(x)$ with rational coefficients.
  • What requirements would need to be placed on the sequence ${x_n}$ to guarantee that $R(x_n) \rightarrow R(x)$ if $x_n \rightarrow x$ and $R(x)$ is some rational function.

2.2.8

Provide a direct proof of iii) in Theorem 14 using the definition of convergence (an $\varepsilon - N$ argument) without using i) or ii) from the algebraic limit rules. i) ii) iii) If $b \neq 0, \exists N \in \mathbb N$ such that for $n > N, b_n \neq 0$ Also it then follows that $\frac{1}{b_n} \rightarrow \frac1b$ and $\frac{a_n}{b_n} \rightarrow \frac{a}b$

2.3.1

Consider the sequence $u_n = (-1)^n$. Write out the first 5 terms of the subsequence ${u_{3k+1}}_{k \geq 1}$

2.3.2

For the sequence ${x_n} = {\sin(\frac{n\pi}4) + \frac1n}$, please write out in simplified form, the following subsequences.

  • ${x_{4n}}$
  • ${x_{4n+1}}$
  • ${x_{4n+2}}$
  • ${x_{4n+3}}$

Which of these subsequences converge? Which diverge?

2.3.3

  • Suppose a subsequence $u_n$ is such that $\lim_{k\rightarrow \infty} u_{2k} = L \in \mathbb Q$ and $\lim_{k \rightarrow \infty} u_{2k+1} = L$. Prove that $\lim_{k \rightarrow \infty} u_k = L$.
  • Prove or disprove. If a sequence is such that $\lim_{k \rightarrow \infty} u_{3k} = Ln \in \mathbb Q$ and $\lim_{k \rightarrow \infty} u_{3k+1} = L$, does this imply that $\lim_{n \rightarrow \infty} u_n = L$?

2.3.4

Given a subsequence ${x_n}$, say you have $M$ distinct subsequences of ${x_n}$ in that none of the terms of the subsequences overlap. Call $A_k$ the indices of subsequence $k$. If every one of these distinct $M$ subsequences converge to the same value $L$, what condition is required of

\[\bigcup_{k=1}^M A_k\]

to guarantee the original sequence ${x_n}$ converges to $L$.