10.1 Sequences Tues Feb 16 Do Now Find the next three terms 1, -½, ¼, -1/8, …

Quiz Review

Sequences A sequence is an ordered collection of numbers defined by a function f(n) on a set o integers The values of f(n) are called the terms of the sequence N is called the index We refer to the nth term as the general term

Ex – Recursive Sequence A recursive sequence is one where the nth term is determined by previous terms

Limit of a Sequence We say that a sequence converges to a limit L if, for every, there is a number M s.t. for all n > M -If no limit exists, the sequence diverges -If the terms increase without bound, the sequence diverges to infinity

Sequence defined by a Function If exists, then the sequence converges to the same limit We can use our old limit rules

Ex Find the limit of the sequence

Ex Calculate

Limit Laws for Sequences Assuming that both sequences converge, the following limit laws apply Sum and Difference Product and Quotient Coefficient

Compositions If f(x) is continuous and then, We can bring a limit inside of a function Remember those limits that use e^x and ln x?

Ex if time Determine the limit of

Bounded Sequences A sequence is: Bounded from above if there is a number M such that every term in the sequence is <= M Bounded from below if there is a number m such that every term is >= m Bounded if it is bounded from above and below Unbounded if it is not bounded at all

Notes: This means that all convergent sequences are bounded We can determine if a sequence converges if it is both bounded and monotonic – Monotonic sequences increase for all n or decrease for all n. They do not do both

Bounded Monotonic Sequences If a sequence is both monotonic and bounded, then it will converge somewhere in between the bounds

Closure Determine the limit of the sequence HW: p.546 #