# Determine whether the following sequences are increasing or decreasing or neither.

Determine whether the following sequences are increasing or decreasing or neither.

a) $$\left(\ n + \sin (n) \right)_{n=0}^\infty$$ b) $$\left(\ n + \sin (1.047n) \right)_{n=1}^\infty$$

For: a) Let $f(x) = x +\sin(x) \implies f'(x) = 1 + \cos(x)$

$$1 + \cos(x)\geq 0$$

and we also know that $$-1 \leq \cos(x) \leq 1$$

But since n starts at 1, the answer will be greater or equal to zero, regarding what n we take, therefore the function is increasing but i have no ideea how to prove that that the sequence is aswell. I know that i have to show that $$\ a(n) \leq a(n+1)$$ but i don't exactly know how. Could you help me ?

• for all rational n, sin n >-1 Mar 20, 2016 at 21:47
• "therefore the function is increasing but i have no ideea how to prove that that the sequence" The sequence is simply $a_n = f(n); n \in \mathbb Z$ If the function is increasing, the sequence, which is simply indexed values of the function is as well. Mar 20, 2016 at 21:57

By the Mean Value Theorem, $\sin(n+1) - \sin(n) = \cos(t)$ for some $t$ between $n$ and $n+1$.
$$\sin((n+1)c) - \sin(nc) = (\cos(c) -1) \sin(nc) + \sin(c) \cos(nc)$$ so by Cauchy-Schwarz $$|\sin((n+1)c) - \sin(nc)| \le \sqrt{(\cos(c)-1)^2 + \sin(c)^2} = \sqrt{2 - 2 \cos(c)}$$