1. The Comparison Test Let 0 a k b k for all k.

2. Mika Seppälä The Comparison Test Comparison Theorem A Assume that 0 a k b k for all k. If the series converges, then also the series converges, and

3. Mika Seppälä The Comparison Test Claim converges. Observe that the partial sums S m = a 1 + a 2 + … + a m form an increasing sequence since a k 0 for all k. Proof

4. Mika Seppälä The Comparison TEST Claim converges. The assumptions imply Observe that the sum is finite since this series converges. Proof (contd)

5. Mika Seppälä The Comparison test Claim converges. Proof (contd) The partial sums form a bounded increasing sequence. Hence the limit exists and is finite.

6. Mika Seppälä The Comparison test Comparison Theorem B Assume that 0 a k b k for all k. If the series diverges, then also the series diverges.

7. Mika Seppälä The Comparison Test Claim diverges. Since the series diverges, the partial sums S m form an unbounded set. The assumptions imply Proof

8. Mika Seppälä Claim diverges. Hence Proof (contd) Since, also THE COMPARISON TEST

9. Mika Seppälä Example Show that the series converges. For all integer values of k, 1 < 2 + sin k < 3. Solution Hence for all integer values of k. THE COMPARISON TEST

10. Mika Seppälä Example Show that the series converges. The series is a convergent geometric series Solution (contd) Hence converges by the Comparison Theorem A. THE COMPARISON TEST

11. Mika Seppälä Example Show that the series diverges. Hence for all positive integer values of k. For all positive integers k, Solution THE COMPARISON TEST

12. Mika Seppälä Example Show that the series diverges. Since the Harmonic Series diverges, also Solution (contd) diverges by the Comparison Theorem B. THE COMPARISON TEST

13. The Comparison Test Let 0 a k b k for all k.