Sequences and Series

Sequences and Series Unit 12

Arithmetic and Geometric Sequences Unit 12: Sequences and Series

Vocabulary

Arithmetic Sequences

Geometric Sequences

Series Unit 12: Sequences and Series

Series

Sigma Notation

Series Shortcuts

Limits of Functions Unit 12: Sequences and Series

Informal Definition of a Limit • Let f be a function and c be a real number such that f(x) is defined for all values of x near x=c. • Whenever x takes on values closer and closer but not equal to c (on both sides of c), the corresponding values of f(x) get very close to, and possibly equal, to the same real number Land the values of f(x) can be made arbitrarily close to L by taking values of x close enough to c, but not equal to c.

Definition of a Limit • The limit of the function f(x) as x approaches c is the number L. • This can be written as:

Examples 3 • Find • Notice that

Examples 1 • Find • Notice that undefined

Examples ∞ • Find • Notice that

When Limits Do Not Exist • If 𝑓(𝑥) approaches ∞ as x approaches c from the right and 𝑓(𝑥) approaches −∞ as x approaches c from the left or 𝑓(𝑥) approaches −∞ as x approaches c from the right and 𝑓(𝑥) approaches ∞ as x approaches c from the left. • Find Does Not Exist

When Limits Do Not Exist • If approaches L as x approaches c from the right and approaches M, with , as x approaches c from the left. • Find Does Not Exist

When Limits Do Not Exist • If 𝑓(𝑥) oscillates infinitely many times between two numbers as x approaches c from either side. • Find Does Not Exist

Limits at Infinity • Let be a function that is defined for all for some number a if: • as , • and the values of can be made arbitrarily close to L by taking large enough values of x, • then the limit of as is L, which is written (the limit of a function is a statement about the end behavior)

Examples 6 • Find • Find + 1 1

Examples 0 • Find • Find 0

Infinite Series Unit 12: Sequences and Series

Convergence of a Sequence

Convergence of a Series

Sequences and Series