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L’H ô pital’s Rule. 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.
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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
Explicit to Implicit • Convert the sequence to implicit form. • Given the function , write the implicit form of the sequence .
Implicit to Explicit • Write the sequence in explicit form. • Write the sequence in explicit form.
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. • Otherwise, the sequence diverges.
Rigorous Definition If, for > 0, there is an integer N such that 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 and ak = f (k) for all k 1, then
The Squeeze Theorem Suppose that ak bk ck for all k 1 and that Then