1. L’Hôpital’s Rule

2. 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, …}

3. What is a sequence? A real-valued function defined for positive (or non-negative) integer inputs. {a n }, where a n = n 2 for n = 1, 2, 3, … {a k }, where a k = 1/k for k = 1, 2, 3, … {a j }, where a j = cos((j-1)  /2) for j = 1, 2, 3, …

4. Notation Implicit Form {a 1, a 2, a 3, …} Explicit Forms

5. Explicit to Implicit 1.Convert the sequence to implicit form. 2.Given the function, write the implicit form of the sequence.

6. Implicit to Explicit 1.Write the sequence in explicit form. 2.Write the sequence in explicit form.

7. The Fibonacci Sequence Defined by the rules: F 1 = 1 F 2 = 1 F n+2 = F n + F n+1 Implicit Form: {1, 1, 2, 3, 5, 8, 13, 21, 34, 55, …} Fibonacci Numbers in Nature

8. The Big Question Once again, it’s this: convergence or divergence? –Let {a k } be a sequence and L a real number. If we can make a k as close to L as we like by making k sufficiently large, the sequence is said to converge to L. –Otherwise, the sequence diverges.

9. Rigorous Definition If, for  > 0, there is an integer N such that then the sequence {a k } is said to converge to the real number L (i.e., {a k } has the limit L).

10. Convergence Theorem Let f be a function defined for x  1. If and a k = f (k) for all k  1, then

11. Algebra with Limits

12. The Squeeze Theorem Suppose that a k  b k  c k for all k  1 and that Then