Question

Determine the monotony of the following sequences.

a) Un = 2n+3 / 5n-1 n € N

b) Un = √(n+3) - √(n+1) n € N^*

c) Un = Σ from r=1 to n (1/ n+r) , n € N

Answer 1

$\text{We have that a function is monotone increasing if its derivative is positive and }\\ \text{and monotone decreasing if its derivative is negative. Therefore we find the derivative}\\ \text{of each of the given functions}\\ \text{For the first function $U_n= \frac{2n +3}{5n-1}$, its derivative is}\\ \qquad \qquad \qquad \qquad \qquad 2-\frac{3}{5n^2}\\ \text{which is positive for all $n \in \mathbb{N}$, hence the function is monotone increasing}\\ \text{For the second function $U_n = \sqrt{n+3}-\sqrt{n+1}$, its derivative is}\\ \qquad \qquad \qquad \qquad \qquad \frac{1}{2\sqrt{n+3}}-\frac{1}{2\sqrt{n+1}}\\ \text{which is negative for all $n \in \mathbb{N}$, hence the function is monotone decreasing}\\ \text{For the third function $U_n = \sum^n_{r=1}\frac{1}{n+r}$, to check for the monotonicity, we compute}\\\text{ $U_n-U_{n-1}= \frac{1}{n+1}-\frac{1}{2n+1}-\frac{1}{2n+2}$ }\\ =\frac{1}{2n+2}-\frac{1}{2n+1}\\ =\frac{-1}{(2n+2)(2n+1)}<0\\ \text{Since $U_n-U_{n-1}<0 \implies U_n < U_{n-1}$, the fuction is monotone decreasing}$