Q. 64 - Calculus

Question

Prove that the sequence of factorials {k!} dominates every sequence of exponential functions {b k}, where b > 0, by applying the ratio test from Theorem 7.6 to the sequence of quotients  $\frac{k!}{bk}$

Step-by-Step Solution

Answer

It is proved that the sequence of factorials {k!} dominates every sequence of exponential functions {b k} 

Step 1. Given information

$\frac{k!}{bk}$

Step 2. Proof

$A_k=\frac{k!}{b^k}$

Therefore,

 $\begin{array}{r}\frac{A_{k+1}}{A_k}=\frac{\frac{(k+1)!}{b^{k+1}}}{{\displaystyle \frac{k!}{b^k}}}\\ \\ =\frac{(k+1)!}{b^{k+1}}\cdot \frac{b^k}{k!}\\ =\frac{(k+1)k!}{b^{k}b}\cdot \frac{b^k}{k!}\\ =\frac{(k+1)}{b}\end{array}$

Sequence of quotients $\left\{A_k\right\}=\left\{\frac{k!}{b^k}\right\}$ is an increasing sequence.

Step 3. Calculating the limit

Thus, sequence of factorials {k!} dominates every sequence of exponential functions.