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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 Chapter 10
10.1 Sequences • A sequence can be thought as a list of numbers written in a definite order
Limit of a sequence • Consider the sequence • If we plot some values we get this graph
Limit of a sequence • Consider the sequence
Limit of a sequence • Since a sequence is a collection of numbers, we could have a random collection
Limit of a sequence • Consider the Fibonacci sequence
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
Limit of a sequence (Definition 2) • A sequence has the limit if for every there is a corresponding integer N such that • We write
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
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
The limit laws • If and are convergent sequences and c is a constant, then
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
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
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.
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
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
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.
Laws of Series • If and both converge, then • Note that the laws do not apply to multiplication, division nor exponentiation.
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
Proof Divergence Test • If , then
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.
Geometric Series If we multiply both sides by r we get If we subtract (2) from (1), we get
Geometric Series • An infinite GS diverges if , otherwise
Telescoping Series To solve we will use the identity:
Harmonic Series • Basically this implies that TOO BIG!!!
P-Series • A p-series is a series of the form • Convergence of p-series:
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
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
Absolute/Conditional Convergence • is called absolutely convergent if converges • Absolute convergence theorem: • If convs. Also convs. • (In words) if convs. Abs. convs.
