# 4.E: Convergence of Sequences and Series (Exercises)

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$

( \newcommand{\kernel}{\mathrm{null}\,}\) $$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\id}{\mathrm{id}}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\kernel}{\mathrm{null}\,}$$

$$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$

$$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$

$$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

$$\newcommand{\vectorA}[1]{\vec{#1}} % arrow$$

$$\newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$$

$$\newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vectorC}[1]{\textbf{#1}}$$

$$\newcommand{\vectorD}[1]{\overrightarrow{#1}}$$

$$\newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}}$$

$$\newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}}$$

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

$$\newcommand{\avec}{\mathbf a}$$ $$\newcommand{\bvec}{\mathbf b}$$ $$\newcommand{\cvec}{\mathbf c}$$ $$\newcommand{\dvec}{\mathbf d}$$ $$\newcommand{\dtil}{\widetilde{\mathbf d}}$$ $$\newcommand{\evec}{\mathbf e}$$ $$\newcommand{\fvec}{\mathbf f}$$ $$\newcommand{\nvec}{\mathbf n}$$ $$\newcommand{\pvec}{\mathbf p}$$ $$\newcommand{\qvec}{\mathbf q}$$ $$\newcommand{\svec}{\mathbf s}$$ $$\newcommand{\tvec}{\mathbf t}$$ $$\newcommand{\uvec}{\mathbf u}$$ $$\newcommand{\vvec}{\mathbf v}$$ $$\newcommand{\wvec}{\mathbf w}$$ $$\newcommand{\xvec}{\mathbf x}$$ $$\newcommand{\yvec}{\mathbf y}$$ $$\newcommand{\zvec}{\mathbf z}$$ $$\newcommand{\rvec}{\mathbf r}$$ $$\newcommand{\mvec}{\mathbf m}$$ $$\newcommand{\zerovec}{\mathbf 0}$$ $$\newcommand{\onevec}{\mathbf 1}$$ $$\newcommand{\real}{\mathbb R}$$ $$\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}$$ $$\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}$$ $$\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}$$ $$\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}$$ $$\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$$ $$\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$$ $$\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$$ $$\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$$ $$\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}$$ $$\newcommand{\laspan}[1]{\text{Span}\{#1\}}$$ $$\newcommand{\bcal}{\cal B}$$ $$\newcommand{\ccal}{\cal C}$$ $$\newcommand{\scal}{\cal S}$$ $$\newcommand{\wcal}{\cal W}$$ $$\newcommand{\ecal}{\cal E}$$ $$\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}$$ $$\newcommand{\gray}[1]{\color{gray}{#1}}$$ $$\newcommand{\lgray}[1]{\color{lightgray}{#1}}$$ $$\newcommand{\rank}{\operatorname{rank}}$$ $$\newcommand{\row}{\text{Row}}$$ $$\newcommand{\col}{\text{Col}}$$ $$\renewcommand{\row}{\text{Row}}$$ $$\newcommand{\nul}{\text{Nul}}$$ $$\newcommand{\var}{\text{Var}}$$ $$\newcommand{\corr}{\text{corr}}$$ $$\newcommand{\len}[1]{\left|#1\right|}$$ $$\newcommand{\bbar}{\overline{\bvec}}$$ $$\newcommand{\bhat}{\widehat{\bvec}}$$ $$\newcommand{\bperp}{\bvec^\perp}$$ $$\newcommand{\xhat}{\widehat{\xvec}}$$ $$\newcommand{\vhat}{\widehat{\vvec}}$$ $$\newcommand{\uhat}{\widehat{\uvec}}$$ $$\newcommand{\what}{\widehat{\wvec}}$$ $$\newcommand{\Sighat}{\widehat{\Sigma}}$$ $$\newcommand{\lt}{<}$$ $$\newcommand{\gt}{>}$$ $$\newcommand{\amp}{&}$$ $$\definecolor{fillinmathshade}{gray}{0.9}$$

## Q1

Prove that if $$\lim_{n \to \infty }s_n = s$$ then $$\lim_{n \to \infty }\left |s_n \right | = \left |s \right |$$. Prove that the converse is true when $$s = 0$$, but it is not necessarily true otherwise.

## Q2

1. Let $$(s_n)$$ and $$(t_n)$$ be sequences with $$s_n ≤ t_n,∀n$$. Suppose $$\lim_{n \to \infty }s_n = s$$ and $$\lim_{n \to \infty }t_n = t$$. Prove $$s ≤ t$$. [Hint: Assume for contradiction, that $$s > t$$ and use the definition of convergence with $$ε = \(frac{s-t}{2}$$ to produce an $$n$$ with $$s_n > t_n$$.]
2. Prove that if a sequence converges, then its limit is unique. That is, prove that if $$\lim_{n \to \infty }s_n = s$$ and $$\lim_{n \to \infty }s_n = s$$, then $$s = t$$.

## Q3

Prove that if the sequence $$(s_n)$$ is bounded then $$\lim_{n \to \infty }\left (\frac{s_n}{n} \right ) = 0$$.

## Q4

1. Prove that if $$x \neq 1$$, then $1 + x + x^2 +\cdots + x^n = \frac{1 - x^{n+1}}{1-x}$
2. Use (a) to prove that if $$|x| < 1$$, then $$\lim_{n \to \infty }\left ( \sum_{j=0}^{n} x^j \right ) = \frac{1}{1-x}$$

## Q5

Prove $\lim_{n \to \infty }\frac{a_0 + a_1n + a_2n^2 +\cdots + a_kn^k}{b_0 + b_1n + b_2n^2 +\cdots + b_kn^k} = \frac{a_k}{b_k}$

provided $$b_k \neq 0$$. [Notice that since a polynomial only has finitely many roots, then the denominator will be non-zero when n is sufficiently large.]

## Q6

Prove that if $$\lim_{n \to \infty }s_n = s$$ and $$\lim_{n \to \infty }(s_n - t_n) = 0$$, then $$\lim_{n \to \infty }t_n = s$$.

## Q7

1. Prove that if $$\lim_{n \to \infty }s_n = s$$ and $$s < t$$, then there exists a real number $$N$$ such that if $$n > N$$ then $$s_n < t$$.
2. Prove that if $$\lim_{n \to \infty }s_n = s$$ and $$r < s$$, then there exists a real number $$M$$ such that if $$n > M$$ then $$r < s_n$$.

## Q8

Suppose $$(s_n)$$ is a sequence of positive numbers such that $$\lim_{n \to \infty }\left ( \frac{s_{n+1}}{s_n} \right ) = L$$

1. Prove that if $$L < 1$$, then $$\lim_{n \to \infty }s_n = 0$$. [Hint: Choose $$R$$ with $$L < R < 1$$. By the previous problem, $$∃\; N$$ such that if $$n > N$$, then $$\frac{s_{n+1}}{s_n} < R$$. Let $$n_0 > N$$ be fixed and show $$s_{n_0+k} < R^ks_{n_0}$$. Conclude that $$\lim_{k \to \infty }s_{n_0+k} = 0$$ and let $$n = n_0 + k$$.]
2. Let $$c$$ be a positive real number. Prove $\lim_{n \to \infty }\left ( \frac{c^n}{n!} \right ) = 0$

This page titled 4.E: Convergence of Sequences and Series (Exercises) is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Eugene Boman and Robert Rogers (OpenSUNY) via source content that was edited to the style and standards of the LibreTexts platform.