Q. 15 - Calculus

Question

What does it mean for a sequence $\left\{a_k\right\}$ to be bounded above? Bounded below? Bounded?

Step-by-Step Solution

Answer

If the sequence $\left\{a_k\right\}$ and a real number $M$ such that $M>a_k$ for every $k$, then the sequence is bounded above.

If the sequence $\left\{a_k\right\}$ and a real number $M$ such that $M<a_k$ for every $k$, then the sequence is bounded below.

The sequence which is both bounded above and bounded below is called a bounded sequence.

Step 1 . Given information

The given sequence is, $\left\{a_k\right\}$.

Step 2 . Consider the sequence a k ,

And a real number $M$ such that $M>a_k$ for every $k$, then the sequence is bounded above.

Consider the sequence ${\left\{-n^2\right\}}_{n=0}^{\infty }$

The terms of the sequence are, $\left\{0,-1,-4,....\right\}$.

The sequence is bounded above by $0$.

Hence, the sequence ${\left\{-n^2\right\}}_{n=0}^{\infty }$ is bounded above.

Step 3 . Consider the sequence a k ,

And a real number $M$ such that $M<a_k$ for every $k$, then the sequence is bounded below.

Consider the sequence ${\left\{2^n\right\}}_{n=0}^{\infty }$

The terms of the sequence are, $\left\{1,4,8,...\right\}$.

The sequence is bounded above by $1.$

Hence, the sequence ${\left\{2^n\right\}}_{n=0}^{\infty }$ is bounded below.

Step 4 . The sequence which is both bounded above and bounded below is called a bounded sequence.

Consider the sequence ${\left\{{\left(-1\right)}^{n+1}\right\}}_{n=1}^{\infty }$

The terms of the sequence are, $\left\{-1,1,-1,1,....\right\}$.

The sequence is bounded below by $-1$ and bounded above by $1.$

Hence, the sequence ${\left\{{\left(-1\right)}^{n+1}\right\}}_{n=1}^{\infty }$ is a bounded sequence.