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L’Hôpital’s Rule Let f and g be differenti able, such that a) as x a, either i) f x 0 and g x 0; or ii) f x and g x ; f x b) lim exists. xa g x Then f x f x lim lim . x a g x x a g x What is a sequence? • An infinite, ordered list of numbers. {1, 4, 9, 16, 25, …} {1, 1/2, 1/3, 1/4, 1/5, …} {1, 0, 1, 0, 1, 0, –1, 0, …} What is a sequence? • A real-valued function defined for positive (or non-negative) integer inputs. {an}, where an= n2 for n = 1, 2, 3, … {ak}, where ak= 1/k for k = 1, 2, 3, … {aj}, where aj= cos((j-1)/2) for j = 1, 2, 3, … Notation • Implicit Form {a1, a2, a3, …} • Explicit Forms an a n 1 an n1 Explicit to Implicit 1. Convert the sequence 2. 1 n to 2 0 implicit form. 2x 1 Given the function f x 3 , write the x implicit form of the sequence f nn1. Implicit to Explicit 1. Write the sequence form. 2. Write the sequence explicit form. 1 1 1 1 , , , , in 3 9 27 explicit 1 1 1 1 , , , , in 2 4 8 16 The Fibonacci Sequence • Defined by the rules: F1 = 1 F2 = 1 Fn+2 = Fn + Fn+1 • Implicit Form: {1, 1, 2, 3, 5, 8, 13, 21, 34, 55, …} • Fibonacci Numbers in Nature The Big Question • Once again, it’s this: convergence or divergence? – Let {ak} be a sequence and L a real number. If we can make ak as close to L as we like by making k sufficiently large, the sequence is said to converge to L. lim ak L or ak L k – Otherwise, the sequence diverges. Rigorous Definition If, for > 0, there is an integer N such that ak L k N then the sequence {ak} is said to converge to the real number L (i.e., {ak} has the limit L). Convergence Theorem Let f be a function defined for x 1. If lim f x L x and ak = f (k) for all k 1, then lim ak L. k Algebra with Limits If lim an A and lim bn B then n n 1) lim can cA n 2) lim an bn A B n 3) lim an bn A B n 4) lim anbn AB n an A 5) lim , provided B 0. n b B n The Squeeze Theorem Suppose that ak bk ck for all k 1 and that lim ak lim ck L. k k Then lim bk L. k