Q. 62 - Calculus

Question

Let r > 1.

(a) Show that the series $\sum _{k=1}^{\infty }\frac{r^k}{k!}$ converges.

(b) Explain why part (a) proves that$\underset{k\to \infty }{\lim }\frac{r^k}{k!}=0.$

(c) Explain why part (b) proves that factorial growth dominates exponential growth.

Step-by-Step Solution

Answer

Part a. The given series converges.

Part b. It proved that $\underset{k\to \infty }{\lim }\frac{r^k}{k!}=0.$

Part c. Exponential growth dominates polynomial growth because as $k\to \infty$ the factorial function $k!$ will increase more than exponential growth $r^k.$ Thus, by the definition of dominance, factorial growth dominates exponential growth.

Part (a) Step 1. Given Information.

The given series is $\sum _{k=1}^{\infty }\frac{r^k}{k!}.$

Part (a) Step 2. Showing that the given series converges.

To show that the given series converges we will use the ratio test.

Let the general term is $a_k=\frac{r^k}{k!}.$

So, $a_{k+1}=\frac{r^{k+1}}{\left(k+1\right)!}.$

Now,

$$\begin{gathered} \rho =\underset{k\to \infty }{\lim }\frac{a_{k+1}}{a_k} \\ \rho =\underset{k\to \infty }{\lim }\frac{{\displaystyle \frac{r^{k+1}}{\left(k+1\right)!}}}{{\displaystyle \frac{r^k}{k!}}} \\ \rho =\underset{k\to \infty }{\lim }\frac{\left(k!\right)r^{k+1}}{\left(k+1\right)!r^k} \\ \rho =\underset{k\to \infty }{\lim }\frac{k!\left(r^k·r\right)}{(k+1)k!r^k} \\ \rho =\underset{k\to \infty }{\lim }\frac{r}{k+1} \\ \rho =r\underset{k\to \infty }{\lim }\frac{1}{k+1} \\ \rho =r\left(0\right) \\ \rho =0 \end{gathered}$$

Since $0<1,$ thus the given series converges.

Part (b) Step 1. Explaining.

To prove that $\underset{k\to \infty }{\lim }\frac{r^k}{k!}=0$ we will use part (a), as we have shown in part (a) that $\sum _{k=1}^{\infty }\frac{r^k}{k!}=0$ converges, so if the series converges then $\underset{k\to \infty }{\lim }\frac{r^k}{k!}=0.$

Hence proved.

Part (c) Step 1. Explaining.

To prove that part (b) proves that factorial growth dominates exponential growth. We will let the exponential function as $r^k.$

Now, as $k\to \infty$ the factorial function $k!$ will increase more than exponential growth $r^k.$

Thus, by the definition of dominance, factorial growth dominates exponential growth.

Hence proved.