1. Ch 9.5 Testing Convergence at Endpoints
Calculus Graphical, Numerical, Algebraic by
Finney, Demana, Waits, Kennedy

7. Convergence of Two Series
What does the ratio test show about convergence of both series?
Use improper integrals to show the area of both curves over the interval 1 ≤ x ≤ ∞.
How does this relate to the ratio test?

8. Convergence of Two Series
1. What does the ratio test show about convergence of both series?
2. Use improper integrals to show the area of both curves over the interval 1 ≤ x ≤ ∞.
3. How does this relate to the ratio test? Ratio Test is inconclusive when L = 1; but Integral Test works.

10. Using the Ratio Test gives a limit L =1 which is inconclusive.

12. The p-Series Test

13. The p-Series Test

14. Slow Divergence of Harmonic Series

15. Example

16. Example

19. Limit Comparison Test

20. Limit Comparison Test

21. Limit Comparison Test

22. Limit Comparison Test

25. Alternating Harmonic Series
Prove that the alternating harmonic series is convergent, but not absolutely convergent. Find a bound for the truncation error after 99 terms.

26. Alternating Harmonic Series

29. Rearranging Alternating Harmonic Series

33. Word of Caution
Although we can use the tests we have developed to find where a given power series converges, it does not tell us what function that power series is converging to. That is why it is so important to estimate the error.

34. Maclaurin Series of a Strange Function
Construct the Maclaurin series for f.
For what values of x does this series converge?
Find all values of x for which the series actually converges to f(x).

35. Maclaurin Series of a Strange Function
Construct the Maclaurin series for f.
For what values of x does this series converge?
The series converges to 0 for all values of x.
3. Find all values of x for which the series actually converges to f(x). The only place that this series actually converges to its f-value is at x = 0

36. Converges to a/(1 - r) if |r| < 1. Diverges if |r| ≥ 1.
Series Diverges
Is lim an = 0?
nth-Term Test no
Geometric
Series Test
Is Σ an = a + ar + ar2 + …?
Converges to a/(1 - r) if |r| < 1.
Diverges if |r| ≥ 1.
p-series Test
Does the series have the form
Series converges if p > 1
Series diverges if p ≤ 1
yes
Absolute
convergence
Does Σ |an| converge? Apply 1
of the Comparison tests, Integral
Test, Ratio Test or nth-Root Test
Original series converges
Alternating
Is Σ an = u1 – u2 + u3 - …?
Is there an integer N such that
un ≥ un+1 ≥… ?
Series converges if un →0.
Otherwise series diverges.
Try partial sums