1. Infinite Series (4/4/14)
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?

2. Convergence of Infinite Series
An infinite series an is said to converge to L if the sequence of partial sums {a1, a1+a2, a1+a2+a3, a1+a2+a3+a4, …} converges to L.
Otherwise the series diverges.
Note that this is again exactly analogous to improper integrals.

3. 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?

4. Clicker Question 1 {1, 1+1/2, 1+1/2+1/4, 1+1/2+1/4+1/8,…}
A. converges to 1.
B. converges to 2.
C. converges to some number greater than 2.
D. diverges

5. Clicker Question 2
What are the first four terms of the sequence of partial sums of the series /3 + 1/9 + 1/27+ …?
A. {1, 1, 1, 1}
B. {1, 1/3, 1/9, 1/27}
C. {1, 4/3, 13/9, 40/27}
D. {1, 4/3, 15/9, 42/27}
E. Huh???

6. 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 r + a r 2 + a r 3 + …, where a is a constant and where the constant ratio is r.

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

8. Examples & Assignment for Monday
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.
Do Exercises 1, 2, 15, 17, 20, 23, 27, 31, 33, 39.