Direct Comparison Test

1. Direct Comparison Test If big series converges then small series converges If small series diverges then big series diverges

2. Limit Comparison Test Take the limit of a fraction of a “guessed” series and a given series. Both converge or diverge (0 & infinity are special)

4. Chapter 9(5)Alternating Series TestAbsolute and Conditional Convergence

5. Alternating series contain both positive and negative terms – the signs alternate This is an Alternating Geometric series Alternating Series Test If terms are positive, limit = 0, and terms get smaller, then an alternating series converges

6. Determine convergence or divergence of: Check an > 0 Check Lim = 0 Check terms get smaller The series converges by the Alternating Series Test (AST)

7. Determine convergence or divergence of: Check an > 0 Check Lim = 0 Check terms get smaller This part fails The Alternating Series Test (AST) can not be applied

8. Does the series converge? Check an > 0 Check Lim = 0 Check terms get smaller Series converges by the (AST)

9. Absolute Convergence

10. Describe the convergence of each: 1) Check an > 0 Lim = 0 AST Fails Check Absolute Value 2) Converges by AST Divergent p-series (p < 1) Conditional Convergence

11. 3) 4) Geometric series with r = ½ < 1 Diverges by the nth term test (Lim ≠ 0 then it diverges) Converges 5) 6) Conditional or absolute? p-series with p = ½ < 1 Diverges The series converges p-series with p = ½ < 1 Diverges The series converges conditionally

12. 7) 8) p-series with p=2>1 converges Converges by DCT Larger Denom makes fraction smaller Diverges by the nth term test (Lim ≠ 0 then it diverges) 9) 10) Continuous if n>1 Positive if n>1 Decreasing if n>1 p-series with p = 7/2 > 1 Converges Converges by Integral Test