Q. 18 - Calculus

Question

Give examples of sequences satisfying the given conditions or explain why such an example cannot exist.

What do we mean when we say factorial growth dominates exponential growth? Provide an example illustrating this fact.

Step-by-Step Solution

Answer

The phase factorial growth dominates the exponential growth means that the rate at which factoring expression increases is much faster than the rate at which the exponential expression increases. 

An example is $\left\{A_k\right\}=\left\{\frac{k!}{b^k}\right\}$.

Step 1. Explain when factorial growth dominates exponential growth.

Consider the phrase factoring growth dominates the exponential growth.

The phase factorial growth dominates the exponential growth means that the rate at which factoring expression increases is much faster than the rate at which the exponential expression increases.

Consider the sequence of factorials $\left\{a_k\right\}=\left\{k!\right\}$ dominates the sequence of exponential functions $\left\{b_k\right\}=\left\{b^k\right\}$.

Assume the sequence of quotients $\left\{A_k\right\}=\left\{\frac{k!}{b^k}\right\}$.

The general term of the sequence is $A_k=\frac{k!}{b^k}$.

Step 2. Find the ratio.

The ratio is given below,

$$\begin{gathered} \frac{A_{k+1}}{A_k}=\frac{{\displaystyle \frac{\left(k+1\right)!}{b^{k+1}}}}{{\displaystyle \frac{k!}{b^k}}} \\ =\frac{\left(k+1\right)!}{b^{k+1}}\times \frac{b_k}{k!} \\ =\frac{{\displaystyle k+1}}{{\displaystyle b}} \end{gathered}$$

Which is greater than $1$ (For $$\begin{gathered} k>1, \\ b>0 \end{gathered}$$).

Thus $A_{k+1}<A_k$ and the sequence of quotients $\left\{A_k\right\}=\left\{\frac{k!}{b^k}\right\}$ is an increasing sequence.

Step 3. Find the value of the limit.

The value of $\underset{\mathrm{k}\to \infty }{\lim }\left\{A_k\right\}$ is given below,

$$\begin{gathered} \underset{\mathrm{k}\to \infty }{\lim }\left\{A_k\right\}=\underset{\mathrm{k}\to \infty }{\lim }\left\{\frac{k!}{b^k}\right\} \\ =\infty \end{gathered}$$

Thus, the sequence of factorials $\left\{a_k\right\}=\left\{k!\right\}$ grows at a faster rate than the sequence of exponential function $\left\{b_k\right\}=\left\{b^k\right\}$.

Hence, the sequence of factorials $\left\{a_k\right\}=\left\{k!\right\}$ dominates the sequence of exponential functions $\left\{b_k\right\}=\left\{b^k\right\}$.