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

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 $$\lim_{n \to \infty }\left ( \frac{c^n}{n!} \right ) = 0$$