Question

Use the ratio test to show that the series is convergent.

Thank you!

Answer 1

Step-by-step explanation:

lim(n→∞) |aₙ₊₁ / aₙ|

lim(n→∞) | (4ⁿ⁺¹ / (n+1)!) / (4ⁿ / n!) |

lim(n→∞) | (4ⁿ⁺¹ / (n+1)!) × (n! / 4ⁿ) |

lim(n→∞) | 4 / (n+1) |

0

Since the limit is less than 1, the series converges.

Answer 2

Answer:

The sum  $\displaystyle \sum^{\infty}_{n = 1} \frac{4^n}{n!}$  converges absolutely.

General Formulas and Concepts:
Calculus

Limits

• Limit Rule [Variable Direct Substitution]:                                                    $\displaystyle \lim_{x \to a} x = a$

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.

$\displaystyle \sum^{\infty}_{n = 1} \frac{4^n}{n!}$

Step 2: Find Convergence

1. [Series] Define:                                                                                             $\displaystyle a_n = \frac{4^n}{n!}$
2. [Series] Set up [Ratio Test]:                                                                         $\displaystyle \sum^{\infty}_{n = 1} \frac{4^n}{n!} \rightarrow \lim_{n \to \infty} \bigg| \frac{4^{n + 1}}{(n + 1)!} \cdot \frac{n!}{4^{n}} \bigg|$
3. [Ratio Test] Simplify:                                                                                    $\displaystyle \lim_{n \to \infty} \bigg| \frac{4^{n + 1}}{(n + 1)!} \cdot \frac{n!}{4^{n}} \bigg| = \lim_{n \to \infty} \bigg| \frac{4}{n + 1} \bigg|$
4. [Ratio Test] Evaluate Limit [Limit Rule - Variable Direct Substitution]:     $\displaystyle \displaystyle \lim_{n \to \infty} \bigg| \frac{4^{n + 1}}{(n + 1)!} \cdot \frac{n!}{4^{n}} \bigg| = 0$
5. [Ratio Test] Check Conclusiveness:                                                           $\displaystyle 0 < 1 \ \checkmark$

∴ since 0 is less than 1, the Ratio Test defines the sum  $\displaystyle \sum^{\infty}_{n = 1} \frac{4^n}{n!}$absolutely convergent.

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Learn more about the Ratio Test: https://brainly.com/question/16618162

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

Topic: AP Calculus BC (Calculus I + II)

Unit: Taylor Series