1. IMPROPER INTEGRALS

2. THE COMPARISON TESTS THEOREM: (THE COMPARISON TEST) In the comparison tests the idea is to compare a given series with a series that is known to be convergent or divergent. convg Known Series geometric P-series Determine whether the series converges or diverges. Example:

3. THE COMPARISON TESTS THEOREM: (THE COMPARISON TEST) In the comparison tests the idea is to compare a given series with a series that is known to be convergent or divergent. divg Known Series geometric P-series Determine whether the series converges or diverges. Example:

4. THE COMPARISON TESTS THEOREM: (THE COMPARISON TEST) In the comparison tests the idea is to compare a given series with a series that is known to be convergent or divergent. convg THEOREM: (THE COMPARISON TEST) divg

5. THE INTEGRAL TEST AND ESTIMATES OF SUMS TERM-112

6. THE COMPARISON TESTS In the comparison tests the idea is to compare a given series with a series that is known to be convergent or divergent. THEOREM: (THE COMPARISON TEST) divg Determine whether the series converges or diverges. Example:

7. THE COMPARISON TESTS THEOREM: (THE LIMIT COMPARISON TEST) Determine whether the series converges or diverges. Example: With positive terms both series converge or both diverge.

8. THE COMPARISON TESTS THEOREM: (THE LIMIT COMPARISON TEST) Determine whether the series converges or diverges. Example: With positive terms both series converge or both diverge. Determine whether the series converges or diverges. Example: Notice that in testing many series we find a suitable comparison series by keeping only the highest powers in the numerator and denominator. REMARK: Converge, then convg and divg, then divg and

9. THE INTEGRAL TEST AND ESTIMATES OF SUMS TERM-112

10. THE INTEGRAL TEST AND ESTIMATES OF SUMS TERM-111

11. THE COMPARISON TESTS Remarks: convg

19. THE INTEGRAL TEST AND ESTIMATES OF SUMS TERM-092