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Chapter 7: Q. 5 (page 655)

Dominance Relationships for Sequences: Order the following sequences by dominance when $a > 0 a n d b > 1$
$\{b^{k}\}$

Short Answer

Expert verified

The required order of dominance sequence is $\ln k < < k^{a} < < b^{k} < < k!$

Step by step solution

Step 1. Given information

The given expression is $\{b^{k}\}$

Step 2. Explanation

Use the concept of dominance relationships for simple sequences,

For any real number $a > 0 a n d b > 1$ , the following string of dominance holds.

$\ln k < < k^{a} < < b^{k} < < k!$ for k sufficiently large.

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Most popular questions from this chapter

Let $\sum_{k = 0}^{\infty} c r^{k}$ and $\sum_{k = 0}^{\infty} b v^{k}$ be two convergent geometric series. If b and v are both nonzero, prove that $\sum_{k = 0}^{\infty} \frac{c r^{k}}{b v^{k}}$ is a geometric series. What condition(s) must be met for this series to converge?

What is meant by the remainder $R_{n}$ of a series $\sum_{k = 1}^{\infty} a_{k}$

Let $\sum_{k = 0}^{\infty} c r^{k}$ and $\sum_{k = 0}^{\infty} b v^{k}$ be two convergent geometric series. Prove that $\sum_{k = 0}^{\infty} (c r^{k} . b v^{k})$ converges. If neither c nor b is 0, could the series be $(\sum_{k = 0}^{\infty} c r^{k}) (\sum_{k = 0}^{\infty} b v^{k})$ ?

Let f(x) be a function that is continuous, positive, and decreasing on the interval $[1, \infty)$ such that $\underset{x \to \infty}{l i m} f (x) = α > 0$ , What can the integral tells us about the series $\sum_{k = 1}^{\infty} f (k)$ ?

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Dominance Relationships for Sequences: Order the following sequences by dominance when a>0andb>1bk

Short Answer

Step by step solution

Step 1. Given information

Step 2. Explanation

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Dominance Relationships for Sequences: Order the following sequences by dominance when $a > 0 a n d b > 1$
$\{b^{k}\}$