1. Test Corrections You may correct your test. You will get back 1/3 of the points you lost if you submit correct answers. This work is to be done on your own (or in consultation with me only). Skidmore Honor Code! Corrections should be done on separate sheets, NOT on the original test. Hand both things in, NOT stapled together. Due on Tuesday (11/20) at 5 pm. Office hours on Tuesday 9-11 and 2-5.

2. Clicker Question 1 {1, 1/2, 1, 1/3, 1, 1/4, 1, 1/5, …} – A. converges to 1 – B. converges to 0 – C. converges to ½ – D. diverges

3. Clicker Question 2 {1, 1 + 1/3, 1 + 1/3 + 1/9, 1 + 1/3 + 1/9 + 1/27, …} – A. converges to 1 – B. converges to 2 – C. converges to some number between 1 and 2 – D. diverges – E. we don’t have enough info to tell if it converges or diverges.

4. Infinite Series (11/16/12) We now study a “discrete” analogue of improper integrals, in which we asked if the areas represented by integrals of unbounded regions converged or diverged. These discrete (as opposed to continuous) objects are just sums, but are called series. Exactly as with improper integrals, we can ask if a given series converges or diverges, and if the former, to what?

5. Convergence of Infinite Series An infinite series  a n is said to converge to L if the sequence of partial sums {a 1, a 1 +a 2, a 1 +a 2 +a 3, a 1 +a 2 +a 3 +a 4, …} converges to L. Otherwise the series diverges. Note that this is again exactly analogous to improper integrals.

6. Some simple (?) examples 1 + 1/2 + 1/4 + 1/8 + 1/16 (this is a finite sum, not a series) 1 + 1/2 + 1/4 + 1/8 + 1/16 + … 1 + 1/3 + 1/9 + 1/27 + … 1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + … 1 + 1/4 + 1/9 + 1/16 + 1/25 + … In each case, given a series: – 1. Does it converge or diverge? – 2. If it converges, to what?

7. Geometric Series The first two series on the previous slide are examples of geometric series. A series is called geometric if the ratio of any two adjacent terms stays constant. In the finite sum and the two series examples, the ratios are 1/2, 1/2, and 1/3. Hence a geometric series is one of the form a + a x + a x 2 + a x 3 + …, where a is a constant and where the constant ratio is x.

8. Summing a geometric series Geometric series are very easy to sum up: just multiply the series by 1  x (x = the ratio). Hence the sum of a finite geometric sum which goes up to a x n is a(1  x n+1 )/(1  x) Use this formula to get the sum of the first example. If the ratio x satisfies that |x| < 1, then note that lim n  x n+1 = 0, so the sum on the previous slide becomes simply a / (1 – x).

9. Examples & Assignment for Mon Use this formula to work out the sum of the second and third examples. Use this formula to find the sum of the infinite geometric series 5 – 5/4 + 5/16 – 5/64 + … Calculate Assignment: - Read Section 11.2 as needed. – Do Exercises 1, 2, 5, 15, 17, 20, 23, 27, 31, 33, 39.