Series (i.e., Sums) (3/22/06) As we have seen in many examples, the definite integral represents summing infinitely many quantities which are each infinitely small (i.e., we take the limit as  x goes to 0). Closely related to this are discrete sums, called series. Series can have finitely many terms or infinitely many terms (“Infinite series”).

Some simple (?) examples 1 + 1/2 + 1/4 + 1/8 + 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 + … Two main questions concerning infinite series are: – 1. Does it converge? (Sound familiar?) – 2. If so, to what? (Also sound familiar?)

Geometric Series The first three 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 three 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 the constant ratio is x.

Summing a geometric series Geometric series, whether finite or infinite, are very easy to sum up. To see why just multiply the series by 1  x (where x is the ratio). Hence the sum of a finite series 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.

Convergence of geometric series 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). 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 for Friday Read Section On page 720-1, do Exercises 2, 11, 12, 14, odd, 35, 41, and 42.