Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

1. Limits and Continuity

Introduction to Limits

Calculus

1. Limits and Continuity

Introduction to Limits

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Multiple Choice

Approximate the sum of the series $\sum_{n = 1} \infty \frac{-^{n} 6 n}{n!}$ correct to four decimal places.

0.0000

- 2.3026

- 3.6109

2.7183

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Verified step by step guidance

Step 1: Recognize that the series $ \sum_{n=1}^{\infty} \frac{(-1)^n 6n}{n!} $ is an alternating series. Alternating series often converge quickly, and their sum can be approximated by considering the first few terms until the error is less than the desired precision.

Step 2: Write out the first few terms of the series explicitly. Substitute $ n = 1, 2, 3, \dots $ into the formula $ \frac{(-1)^n 6n}{n!} $ to compute the terms: $ T_1 = \frac{(-1)^1 \cdot 6 \cdot 1}{1!}, T_2 = \frac{(-1)^2 \cdot 6 \cdot 2}{2!}, T_3 = \frac{(-1)^3 \cdot 6 \cdot 3}{3!}, \dots $.

Step 3: Add the terms sequentially, keeping track of the cumulative sum. For alternating series, the error introduced by truncating the series is less than the absolute value of the first omitted term. Continue adding terms until the error is less than $ 0.0001 $ (four decimal places).

Step 4: Verify convergence using the Alternating Series Test. Ensure that the magnitude of the terms $ \frac{6n}{n!} $ decreases monotonically and approaches zero as $ n \to \infty $. This confirms that the series converges and the approximation is valid.

Step 5: Once the sum is approximated to the desired precision, express the result as the sum of the included terms. The final value should be correct to four decimal places, as specified in the problem.