Q. 38 - Calculus

Question

Find the least upper bound of the sequences in Exercises 37–42 

$\left\{\frac{1}{2},\frac{2}{3},\frac{3}{4},\frac{4}{5},\dots \right\}$

Step-by-Step Solution

Answer

The least upper bound is $1.$

Step 1. Given information.

The given sequence is $\left\{\frac{1}{2},\frac{2}{3},\frac{3}{4},\frac{4}{5},\dots \right\}$.

Step 2. Upper limit.

Now, we can write,

$$\begin{gathered} \underset{k\to \infty }{\lim }{a}_k=\underset{k\to \infty }{\lim }\left(\frac{k}{k+1}\right) \\ =\underset{k\to \infty }{\lim }\left(\frac{1}{1+\frac{1}{k}}\right) \\ =\frac{1}{1+0} \\ =1 \end{gathered}$$