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

0. Functions

Introduction to Functions

Calculus

0. Functions

Introduction to Functions

Previous problemNext problem

Struggling with Calculus?

Join thousands of students who trust us to help them ace their exams!Watch the first video

Multiple Choice

Which of the following is a power series representation for the function $f (x) = x^{8}, \arctan (x^{3})$ centered at $x = 0$ ?

\sum_{n = 0}^{\infty} \frac{{(- 1)}^{n} x^{8 + 2 n}}{(2 n + 1)}

\sum_{n = 0}^{\infty} \frac{{(- 1)}^{n} x^{8 + 6 n}}{(2 n + 1)}

\sum_{n = 0}^{\infty} \frac{x^{8 + 3 n}}{(n + 1)}

\sum_{n = 0}^{\infty} \frac{{(- 1)}^{n} x^{3 n}}{(2 n + 1)}

Previous problemNext problem

Verified step by step guidance

Step 1: Recall the power series representation for the arctan function. The function arctan(x) can be expressed as a power series: arctan(x) = sum_{n=0}^{∞} ((-1)^n * x^(2n+1)) / (2n+1). This series is centered at x = 0.

Step 2: Substitute x^3 into the arctan(x) series to find the power series representation for arctan(x^3). Replace x with x^3 in the formula: arctan(x^3) = sum_{n=0}^{∞} ((-1)^n * (x^3)^(2n+1)) / (2n+1).

Step 3: Simplify the exponent in the substituted series. When (x^3)^(2n+1) is expanded, it becomes x^(3*(2n+1)) = x^(6n+3). Thus, the series becomes: arctan(x^3) = sum_{n=0}^{∞} ((-1)^n * x^(6n+3)) / (2n+1).

Step 4: Multiply the series for arctan(x^3) by x^8 to account for the given function f(x) = x^8 * arctan(x^3). This results in: f(x) = x^8 * sum_{n=0}^{∞} ((-1)^n * x^(6n+3)) / (2n+1).

Step 5: Combine the x^8 term with the x^(6n+3) term in the series. Using the laws of exponents, x^8 * x^(6n+3) = x^(8+6n+3) = x^(8+6n). Thus, the final power series representation for f(x) is: f(x) = sum_{n=0}^{∞} ((-1)^n * x^(8+6n)) / (2n+1).