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

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?

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.

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

The p-Series Test

The p-Series Test

Slow Divergence of Harmonic Series

Example

Example

Limit Comparison Test

Limit Comparison Test

Limit Comparison Test

Limit Comparison Test

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.

Alternating Harmonic Series

Rearranging Alternating Harmonic Series

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.

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).

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

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