1. 12.8 Power Series. Radius and interval of convergence
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2. 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.

3. SOLUTION

4. 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.

5. 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

6. SOLUTION

7. (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

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

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

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

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

12. 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

13. Use Ratio Test 
Converges only if x = 0

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

15. 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

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