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CALCULUS II

CALCULUS II. Chapter 10. 10.1 Sequences. A sequence can be thought as a list of numbers written in a definite order. Examples. http://www.youtube.com/watch?v=Kxh7yJC9Jr0. Limit of a sequence. Consider the sequence If we plot some values we get this graph. Limit of a sequence.

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CALCULUS II

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  1. CALCULUS II Chapter 10

  2. 10.1 Sequences • A sequence can be thought as a list of numbers written in a definite order

  3. Examples

  4. http://www.youtube.com/watch?v=Kxh7yJC9Jr0

  5. Limit of a sequence • Consider the sequence • If we plot some values we get this graph

  6. Limit of a sequence • Consider the sequence

  7. Limit of a sequence • Since a sequence is a collection of numbers, we could have a random collection

  8. Limit of a sequence • Consider the Fibonacci sequence

  9. Limit of a sequence

  10. Limit of a sequence (Definition 1) • A sequence has the limit if we can make the terms as close as we like by taking n sufficiently large. • We write

  11. Limit of a sequence (Definition 2) • A sequence has the limit if for every there is a corresponding integer N such that • We write

  12. Convergence/Divergence • If exists we say that the sequence converges. • Note that for the sequence to converge, the limit must be finite • If the sequence does not converge we will say that it diverges • Note that a sequence diverges if it approaches to infinity or if the sequence does not approach to anything

  13. Divergence to infinity • means that for every positive number M there is an integer N such that • means that for every positive number M there is an integer N such that

  14. The limit laws • If and are convergent sequences and c is a constant, then

  15. The limit laws

  16. L’Hopital and sequences • Theorem: If and , when n is an integer, then • L’Hopital: Suppose that and are differentiable and that near a. Also suppose that we have an indeterminate form of type . Then

  17. More Theorems • Squeeze thm: Let be sequences such that for some M, for and . Then • Continuity: If is continuous and the limit exists, then • Bounded monotonic sequences converge: if for all n, and

  18. Examples

  19. http://www.youtube.com/watch?v=9K1xx6wfN-U

  20. 10.2 Infinite Series • Is the summation of all elements in a sequence. • Remember the difference: Sequence is a collection of numbers, a Series is its summation.

  21. http://www.youtube.com/watch?v=haK3oC0L_a8

  22. Visual proof of convergence • It seems difficult to understand how it is possible that a sum of infinite numbers could be finite. Let’s see an example

  23. Convergence/Divergence • We say that an infinite series converges if the sum is finite, otherwise we will say that it diverges. • To define properly the concepts of convergence and divergence, we need to introduce the concept of partial sum

  24. Convergence/Divergence • The partial sum is the finite sum of the first terms. • converges to if and we write: • If the sequence of partial sums diverges, we say that diverges.

  25. Laws of Series • If and both converge, then • Note that the laws do not apply to multiplication, division nor exponentiation.

  26. Divergence Test • If does not converge to zero, then diverges. • Note that in many cases we will have sequences that converge to zero but its sum diverges

  27. Proof Divergence Test • If , then

  28. Geometric Series First term multiplied by r Third term multiplied by r Second term multiplied by r Note that in this case we start counting from zero. Technically it doesn’t matter, but we have to be careful because the formula we will use starts always at n=0.

  29. Geometric Series If we multiply both sides by r we get If we subtract (2) from (1), we get

  30. Geometric Series • An infinite GS diverges if , otherwise

  31. Examples (not only GS)

  32. http://www.youtube.com/watch?v=xjmy5hkZccY

  33. http://www.youtube.com/watch?v=C8piSCOdo1Y

  34. Telescoping Series To solve we will use the identity:

  35. Telescoping Series

  36. http://www.youtube.com/watch?v=7tDK_UjdWOs

  37. http://www.youtube.com/watch?v=MDYb5DnRH2c

  38. Harmonic Series • Basically this implies that TOO BIG!!!

  39. http://www.youtube.com/watch?v=0XIqnoJ72CU

  40. P-Series • A p-series is a series of the form • Convergence of p-series:

  41. Examples (not only P-series)

  42. Comparison Test • Assume that there exists such that for • If converges, then also converges. • If diverges, then also diverges. • if diverges this test does not help • Also, if converges this test does not help

  43. Limit Comparison Test • Let and be positive sequences. Assume that the following limit exists • If , then converges if and only if converges. (Note that L can not be infinity) • If and converges, then converges

  44. Examples

  45. http://www.youtube.com/watch?v=xesQnFWw8f8

  46. http://www.youtube.com/watch?v=8eCFY82HkRA

  47. Absolute/Conditional Convergence • is called absolutely convergent if converges • Absolute convergence theorem: • If convs. Also convs. • (In words) if convs. Abs. convs.

  48. http://www.youtube.com/watch?v=6hOeqjoQvNw

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