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

For which values of $x$ does the series $\sum_{n = 1}^{\infty} \frac{x^{n}}{n}$ converge?

For all real numbers

x

For all

x

such that

| x | < 1

For all

x

such that

- 1 < x \leq 1

For all

x

such that

| x | \leq 1

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Verified step by step guidance

Step 1: Recognize that the problem involves determining the convergence of the infinite series $ \sum_{n=1}^{\infty} \frac{x^n}{n} $. Convergence depends on the behavior of the terms $ \frac{x^n}{n} $ as $ n \to \infty $.

Step 2: Recall the root test or ratio test for series convergence. These tests are often used to analyze series involving powers of $ x $. For this series, the ratio test is particularly useful.

Step 3: Apply the ratio test. The ratio test states that a series $ \sum a_n $ converges absolutely if $ \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| < 1 $. Substitute $ a_n = \frac{x^n}{n} $ into the test.

Step 4: Simplify the ratio $ \left| \frac{a_{n+1}}{a_n} \right| = \left| \frac{x^{n+1}/(n+1)}{x^n/n} \right| = \left| x \right| \cdot \frac{n}{n+1} $. As $ n \to \infty $, $ \frac{n}{n+1} \to 1 $, so the ratio becomes $ \left| x \right| $.

Step 5: Conclude that the series converges absolutely if $ \left| x \right| \leq 1 $. However, further analysis shows that the series converges conditionally for $ -1 < x \leq 1 $. This is because the series diverges for $ x = -1 $ due to the alternating harmonic series behavior.