1. Sec 11.3: THE INTEGRAL TEST AND ESTIMATES OF SUMS a continuous, positive, decreasing function on [1, inf) Convergent THEOREM: (Integral Test) Convergent DivergentDinvergent Test the series for convergence or divergence. Example: Test the series for convergence or divergence. Example:

2. Sec 11.3: THE INTEGRAL TEST AND ESTIMATES OF SUMS a continuous, positive, decreasing function on [1, inf) Convergent THEOREM: (Integral Test) Convergent DivergentDinvergent When we use the Integral Test, it is not necessary to start the series or the integral at n = 1. For instance, in testing the series REMARK:

3. Sec 11.3: THE INTEGRAL TEST AND ESTIMATES OF SUMS a continuous, positive, decreasing function on [1, inf) Convergent THEOREM: (Integral Test) Convergent DivergentDinvergent When we use the Integral Test, it is not necessary to start the series or the integral at n = 1. For instance, in testing the series REMARK: Also, it is not necessary that f(x) be always decreasing. What is important is that f(x) be ultimately decreasing, that is, decreasing for larger than some number N. REMARK:

4. Sec 11.3: THE INTEGRAL TEST AND ESTIMATES OF SUMS a continuous, positive, decreasing function on [1, inf) Convergent THEOREM: (Integral Test) Convergent DivergentDinvergent For what values of p is the series convergent? Example: Special Series: 1)Geometric Series 2)Harmonic Series 3)Telescoping Series 4)p-series

5. Sec 11.3: THE INTEGRAL TEST AND ESTIMATES OF SUMS a continuous, positive, decreasing function on [1, inf) Convergent THEOREM: (Integral Test) Convergent DivergentDinvergent For what values of p is the series convergent? Example: P Series:

6. Sec 11.3: THE INTEGRAL TEST AND ESTIMATES OF SUMS For what values of p is the series convergent? Example: P Series: Example: Test the series for convergence or divergence.

7. Sec 11.3: THE INTEGRAL TEST AND ESTIMATES OF SUMS FINAL-081

8. Sec 11.3: THE INTEGRAL TEST AND ESTIMATES OF SUMS a continuous, positive, decreasing function on [1, inf) Convergent THEOREM: (Integral Test) Convergent DivergentDinvergent Integral Test just test if convergent or divergent. But if it is convergent what is the sum?? REMARK: We should not infer from the Integral Test that the sum of the series is equal to the value of the integral. In fact, REMARK:

9. Sec 11.3: THE INTEGRAL TEST AND ESTIMATES OF SUMS We should not infer from the Integral Test that the sum of the series is equal to the value of the integral. In fact, REMARK: ESTIMATING THE SUM OF A SERIES Example: Estimate the sum How accurate is this estimation?

10. Sec 11.3: THE INTEGRAL TEST AND ESTIMATES OF SUMS We should not infer from the Integral Test that the sum of the series is equal to the value of the integral. In fact, REMARK: ESTIMATING THE SUM OF A SERIES

11. Sec 11.3: THE INTEGRAL TEST AND ESTIMATES OF SUMS ESTIMATING THE SUM OF A SERIES REMAINDER ESTIMATE FOR THE INTEGRAL TEST

12. Sec 11.3: THE INTEGRAL TEST AND ESTIMATES OF SUMS REMAINDER ESTIMATE FOR THE INTEGRAL TEST

13. Sec 11.3: THE INTEGRAL TEST AND ESTIMATES OF SUMS

14. TERM-102