Q. 17 - Calculus

Question

Give an example of a sequence with neither an upper bound nor a lower bound. 

Step-by-Step Solution

Answer

Ans:  The sequence ${\left\{(-1{)}^{k}k\right\}}_{k=0}^{\mathrm{\infty }}$ neither has an upper bound nor a lower bound.

Step 1. Given information.

Give an example of a sequence with neither an upper bound nor a lower bound. 

Step 2. The objective is to give an example of the sequence which has neither an upper bound nor a lower bound.

   Consider the sequence ${\left\{(-1{)}^{k}k\right\}}_{k=0}^{\mathrm{\infty }}$

Find the terms of the sequence by substituting $k=0,1,2,3,....$

The terms of the sequence are $\left\{(-1{)}^0\cdot 0,(-1{)}^1\cdot 1,(-1{)}^2\cdot 2,(-1{)}^3\cdot 3,(-1{)}^4\cdot 4,\dots \cdot \right\}$

Hence, the terms of the sequence are  $\{ 0,-1,2,-3,4,... \}$.

Step 3. Thus,

The above sequence is neither an increasing sequence nor a decreasing sequence. 

Hence, the sequence neither has an upper bound nor a lower bound.