Problem 6.4.1.
Prove that if \(\lim_{n\rightarrow\infty}s_n=s\) then \(\lim_{n\rightarrow\infty}|s_n|=|s|\text{.}\) Prove that the converse is true when \(s=0\text{,}\) but it is not necessarily true otherwise.
Section 6.4 Additional Problems
Problem 6.4.2.
(a)
Let \(\left(s_n\right)\) and \(\left(t_n\right)\) be sequences with \(s_n\leq t_n,\forall n\text{.}\) Suppose \(\lim_{n\rightarrow\infty}s_n=s\) and \(\lim_{n\rightarrow\infty}t_n=t\text{.}\)
Prove \(s\leq t\text{.}\)
Hint.
Assume for contradiction, that \(s>t\) and use the definition of convergence with \(\eps=\frac{s-t}{2}\) to produce an \(n\) with \(s_n>t_n\text{.}\)
(b)
Prove that if a sequence converges, then its limit is unique. That is, prove that if \(\lim_{n\rightarrow\infty}s_n=s\) and \(\lim_{n\rightarrow\infty}s_n=t\text{,}\) then \(s=t\text{.}\)
Problem 6.4.3.
Prove that if the sequence \(\left(s_n\right)\) is bounded then \(\lim_{n\rightarrow\infty}\left(\frac{s_n}{n}\right)=0\text{.}\)
Problem 6.4.4.
(a)
Prove that if \(x\neq 1\text{,}\) then
\begin{equation*}
1+x+x^2+\cdots+x^n=\frac{1-x^{n+1}}{1-x}\text{.}
\end{equation*}
(b)
Use (a) to prove that if \(|x|\lt 1\text{,}\) then \(\lim_{n\rightarrow\infty}\left(\sum_{j=0}^nx^j\right)=\frac{1}{1-x}\text{.}\)
Problem 6.4.5.
Prove
\begin{equation*}
\lim_{n\rightarrow\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}\text{,}
\end{equation*}
provided \(b_k\neq 0\text{.}\) [Notice that since a polynomial only has finitely many roots, then the denominator will be non-zero when \(n\) is sufficiently large.]
Problem 6.4.6.
Prove that if \(\lim_{n\rightarrow\infty}s_n=s\) and \(\lim_{n\rightarrow\infty}\left(s_n-t_n\right)=0\text{,}\) then \(\lim_{n\rightarrow\infty}t_n=s\text{.}\)
Problem 6.4.7.
(a)
Prove that if \(\lim_{n\rightarrow\infty}s_n=s\) and \(s\lt
t\text{,}\) then there exists a real number \(N\) such that if \(n>N\) then \(s_n\lt t\text{.}\)
(b)
Prove that if \(\lim_{n\rightarrow\infty}s_n=s\) and \(r\lt
s\text{,}\) then there exists a real number \(M\) such that if \(n>M\) then \(r\lt s_n\text{.}\)
Problem 6.4.8.
Suppose \(\left(s_n\right)\) is a sequence of positive numbers such that
\begin{equation*}
\lim_{n\rightarrow\infty}\left(\frac{s_{n+1}}{s_n}\right)=L\text{.}
\end{equation*}
(a)
Prove that if \(L\lt 1\text{,}\) then \(\lim_{n\rightarrow\infty}s_n=0\text{.}\)
Hint.
Choose \(R\) with \(L\lt R\lt 1\text{.}\) By the previous problem, \(\exists\) \(N\) such that if \(n>N\text{,}\) then \(\frac{s_{n+1}}{s_n}\lt R\text{.}\) Let \(n_0>N\) be fixed and show \(s_{n_0+k}\lt R^ks_{n_0}\text{.}\) Conclude that \(\lim_{k\rightarrow\infty}s_{n_0+k}=0\) and let \(n=n_0+k\text{.}\)
(b)
Let \(c\) be a positive real number. Prove \(\displaystyle\lim_{n\rightarrow\infty}\left(\frac{c^n}{n!}\right)=0\text{.}\)
