Sequences and SeriesUnit 12

Arithmetic and Geometric SequencesUnit 12: Sequences and Series

Vocabulary

Arithmetic Sequences

Geometric Sequences

SeriesUnit 12: Sequences and Series

Series

Sigma Notation

Series Shortcuts

Series Shortcuts

Limits of FunctionsUnit 12: Sequences and Series

Informal Definition of a LimitLet 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 L and 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 LimitThe limit of the function f(x) as x approaches c

is the number L.

This can be written as:

ExamplesFind

Notice that

3

ExamplesFind

Notice that undefined

1

ExamplesFind

Notice that

∞

When Limits Do Not ExistIf 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 ExistIf 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 ExistIf 𝑓(𝑥) oscillates infinitely many times

between two numbers as x approaches c from either side.

Find

Does Not Exist

Limits at InfinityLet 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)

ExamplesFind Find

+ 1

6

1

ExamplesFind Find

0

0

Infinite SeriesUnit 12: Sequences and Series

Convergence of a Sequence

Convergence of a Series

Convergence of a Series