Question

Use the ratio test to determine whether ∑n=14∞n+2n! converges or diverges. (a) Find the ratio of successive terms. Write your answer as a fully simplified fraction. For n≥14, limn→∞∣∣∣an+1an∣∣∣=limn→∞.

Answer 1

Answer:

The sum  $\displaystyle \sum^{\infty}_{n = 14} n + 2n!$  diverges ∵ of the Ratio Test.

General Formulas and Concepts:
Calculus

Limits

• Limit Rule [Variable Direct Substitution]:                                                       $\displaystyle \lim_{x \to c} x = c$
• Special Limit Rule [Coefficient Power Method]:                                      $\displaystyle \lim_{x \to \pm \infty} \frac{ax^n}{bx^n} = \frac{a}{b}$

Series Convergence Tests

• Ratio Test:                                                                                                        $\displaystyle \lim_{n \to \infty} \bigg| \frac{a_{n + 1}}{a_n} \bigg|$

Step-by-step explanation:

Step 1: Define

Identify.

$\displaystyle \sum^{\infty}_{n = 14} n + 2n!$

Step 2: Find Convergence

1. [Series] Define:                                                                                             $\displaystyle a_n = n + 2n!$
2. [Series] Set up [Ratio Test]:                                                                         $\displaystyle \sum^{\infty}_{n = 14} n + 2n! \rightarrow \lim_{n \to \infty} \bigg| \frac{n + 1 + 2(n + 1)!}{n + 2n!} \bigg|$
3. [Ratio Test] Evaluate Limit [Coefficient Power Method]:                            $\displaystyle \lim_{n \to \infty} \bigg| \frac{n + 1 + 2(n + 1)!}{n + 2n!} \bigg| = \infty$
4. [Ratio Test] Define conclusiveness:                                                           $\displaystyle \infty > 1$

Since infinity is greater than 1, the Ratio Test defines the sum  $\displaystyle \sum^{\infty}_{n = 14} n + 2n!$  to be divergent.

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Learn more about the Ratio Test: https://brainly.com/question/16654521

Learn more about Taylor Series: https://brainly.com/question/23558817

Topic: AP Calculus BC (Calculus I + II)

Unit: Taylor Series