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Infinite Series (4/4/14)

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 .

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Infinite Series (4/4/14)

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  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 1 + 1/3 + 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 forma + a r + ar 2 + ar 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 ar 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.

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