SEQUENCES DEF: A sequence is a list of numbers in a given order: Example first term second term n-th term index Example Example

SEQUENCES Find the n-term Example Example Example Example Example Even numbers Example odd numbers Example Example

SEQUENCES Find the n-term TERM-132                      

SEQUENCES Find the n-term TERM-092 TERM-131

Recursive Definitions SEQUENCES Recursive Definitions

Recursive Definitions SEQUENCES Example Find a formula for the general term of the sequence This sequence arose when the 13th-century Italian mathematician known as Fibonacci Example Find a formula for the general term of the sequence

SEQUENCES Representation

SEQUENCES Representing Sequences Example

SEQUENCES Limit of a Sequence

SEQUENCES Example Example LIMIT OF THE SEQUENCE as convg Even numbers What is the last number in the sequence ? Example What is the last number in the sequence ? LIMIT OF THE SEQUENCE as We say the sequence convg

SEQUENCES LIMIT OF THE SEQUENCE as convg Remark: Remark: We say the sequence convg If converges to L, we write Remark: If there exist no L then we say the sequence is divergent. Remark: or simply and call L the limit of the sequence

1 2 3 SEQUENCES Convergence or Divergence How to find a limit of a sequence Example (IF you can) use Math-101 to find the limit. Use other prop. To find the limit 1 Example: 2 1)Sandwich Thm: 2)Cont. Func. Thm: 3 3)L’Hôpital’s Rule:

SEQUENCES

SEQUENCES TERM-132  

SEQUENCES Final-142 Final-141

SEQUENCES

SEQUENCES How to find a limit of a sequence You find it in a sequence but not in a function (IF you can) use Math-101 to find the limit. Use other prop. To find the limit Example: 1)Sandwich Thm: 2)Cont. Func. Thm: 3)L’Hôpital’s Rule:

SEQUENCES Factorial; NOTE Example REMARKS

SEQUENCES Example Find where

SEQUENCES

SEQUENCES Example For what values of r is the sequence convergent?

SEQUENCES Example:

Sec 11.1: SEQUENCES Example Note:

SEQUENCES How to find a limit of a sequence (convg or divg) (IF you can) use Math-101 to find the limit. Use other prop. To find the limit abs,r^n,bdd+montone Example: Example: 1)Sandwich Thm: 1)Absolute value: 2)Cont. Func. Thm: 2)Power of r: 3)L’Hôpital’s Rule: 3)bdd+montone: Bdd + monton  convg

SEQUENCES Example Find Faster Example Find where

SEQUENCES Final-122

SEQUENCES

Multiple-Choice Problems SEQUENCES Extra Multiple-Choice Problems

SEQUENCES

SEQUENCES

SEQUENCES

SEQUENCES

SEQUENCES

SEQUENCES Final-141

SEQUENCES TERM-092

SEQUENCES TERM-092

SEQUENCES TERM-082

SEQUENCES If the sequence is convergent, guess the value of the limit and then prove your guess.

SEQUENCES Example: bounded unbounded unbounded bounded If is bounded from above and below, If is not bounded we say that bounded unbounded Example: unbounded bounded

SEQUENCES Example Find where Sol: by sandw. limit is 0

Sec 11.1: SEQUENCES increasing decreasing DEFINITION Example Example Is the sequence increasing or decreasing Example DEFINITION decreasing Is the sequence increasing or decreasing 2-solutions Sol_1 Sol_2 DEFINITION monotonic if it is either increasing or decreasing.

Every bounded, monotonic sequence is convergent Sec 11.1: SEQUENCES DEFINITION bounded above Example if there is a number M such that Is the sequence bounded above or below DEFINITION bounded below if there is a number m such that DEFINITION bounded If it is bounded above and below THEOREM; Every bounded, monotonic sequence is convergent