Question

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

Short answer

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

Step-by-step solution

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 arerole="math" localid="1649305772298" $\{0,-1,2,-3,4,...\}$.

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.