12.8 Power Series. Radius and interval of convergence Mathboat.com

Test endpoints Ratio Test (A) (B) (C) (D) (E) Terms of this alternating series are positive, decreasing and n th term approaches 0. Converges by AST.

SOLUTION

Ratio Test Test endpoints Diverges by BCT Harmonic, Diverges Terms of this alternating series are positive, decreasing and n th term approaches 0. Converges by AST.

diverges. sum doesn’t stop growing Test endpoints (A) (B) (C) (D) (E) Diverges, sum=1 or 0 Ratio Test diverges. sum doesn’t stop growing

SOLUTION

(A) 0 (B) 3 (C) 4 (D) 5 (E) none Test endpoints Eliminate D Terms of this alternating series are positive, decreasing and n th term approaches 0. Converges by AST. Eliminate D

(A) (B) (C) (D) (E) Ratio Test Test endpoints alternating harmonic series converges by AST Harmonic series, diverges

8.What is the radius of convergence of The radius of convergence is

9. What are all the values of x for which the series converges? Check Endpoints

10. What are all values of x for which the series converges? Check Endpoints

Omit negatives in absolute value Check endpoints! Use Ratio Test Omit negatives in absolute value Harmonic divergent Converges by AST; Alternating Harmonic Absolutely Converges if

Use Ratio Test  Converges only if x = 0

Use Ratio Test 0 is always < 1  Converges for all real numbers

Find the Radius of Convergence! Find the Radius of Convergence! Use Ratio Test Check endpoints! Remember: Interval of Convergence is Remember: Interval of Convergence is Converges by AST Find the Radius of Convergence! Find the Radius of Convergence! L’Hopitals Rule Use Basic Comparison Test harmonic divergent diverges too

Find the Radius of Convergence! 15. Interval: Check endpoints! Use Ratio Test Remember: Interval of Convergence is Find the Radius of Convergence! diverges diverges