Question

(3) For every $n \in \mathbb{N}$, denote by $p_n$ the $n$-th prime number, and let $g_n = p_{n+1} - p_n$. (i) Show that $\lim \sup_{n \to \infty} g_n = \infty$. (ii) Let $A = \{n \in \mathbb{N} : n, n+2 \text{ are prime}\}$. Show that $\lim \inf_{n \to \infty} g_n = 2$ if A is infinite, whereas $\lim \inf_{n \to \infty} g_n \ge 4$ if A is finite. Hints: In (i), consider the numbers $n! + j$ for $j = 2, 3, \dots, n$; in (ii), note that $g_n$ is even for all $n \ge 2$.

          (3) For every $n \in \mathbb{N}$, denote by $p_n$ the $n$-th prime number, and let $g_n = p_{n+1} - p_n$.
(i) Show that $\lim \sup_{n \to \infty} g_n = \infty$.
(ii) Let $A = \{n \in \mathbb{N} : n, n+2 \text{ are prime}\}$. Show that $\lim \inf_{n \to \infty} g_n = 2$ if A is infinite, whereas $\lim \inf_{n \to \infty} g_n \ge 4$ if A is finite.
Hints: In (i), consider the numbers $n! + j$ for $j = 2, 3, \dots, n$; in (ii), note that $g_n$ is even for all $n \ge 2$.

(3) For every n ∈ℕ, denote by pn the n-th prime number, and let gn = pn+1 - pn.
(i) Show that limsupn →∞ gn = ∞.
(ii) Let A = {n ∈ℕ : n, n+2 are prime}. Show that liminfn →∞ gn = 2 if A is infinite, whereas liminfn →∞ gn ≥ 4 if A is finite.
Hints: In (i), consider the numbers n! + j for j = 2, 3, …, n; in (ii), note that gn is even for all n ≥ 2.

Added by Douglas N.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Adi S

05/23/2023

Step 1

Step 1: In order to prove (i), we need to show that lim supn→∞ 9n = 0. Show more…

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Texts: (3) For every n ∈ N, denote by Pn the n-th prime number, and let gn = Pn+1 - Pn: (i) Show that lim supn→∞ 9n = 0. (ii) Let A = {n ∈ N : n, n+2 are prime}. Show that lim infn→∞ 9n = 2 if A is infinite, whereas lim infn→∞ 9n = 4 if A is finite. Hints: In (i), consider the numbers n! + j for j = 2, 3, ..., n; in (ii), note that 9n is even for all n > 2.