Sequences

Definition - A function whose domain is the set of all positive integers.

Finite Sequence - finite number of values or elements

2 71, ,6, 2,4,6,8,103 8

Infinite Sequence - infinite number of values or elements

4,7,8,13, 1,3,5,7,9

Notation - n na or b

Section 10.1 - Sequences

Definition - A function whose domain is the set of all positive integers.

Section 10.1 - Sequences

Section 10.1 - Sequences

Three Types of Sequences

Specified – enough information is given to find a pattern 1,4,7,10,13, 2,5,11,23,47,

Explicit Formula

Recursion Formula

𝑎𝑛=3𝑛−2 ,𝑛≥1

𝑏𝑛=𝑏𝑛−1+3 ,𝑛≥2 ,𝑏1=1

Section 10.1 - Sequences

Definitions

If a sequence has a limit that exists, then it is convergent and it converges to the limit value.

If a sequence has a limit that does not exist, then it is divergent.

Theorems Given then implies

If the then

Given then implies

If the then

Section 10.1 - Sequences

Section 10.1 - Sequences

Section 10.1 - Sequences

Section 10.1 - Sequences

Section 10.2 – Infinite Series

Section 10.2 – Infinite Series

Geometric Series

∑𝒏=𝟏

∞

𝒂𝒓𝒏−𝟏=𝒂+𝒂𝒓 +𝒂𝒓𝟐+𝒂𝒓𝟑+⋯𝒂𝒓𝒏−𝟏+𝒂𝒓 𝒏

A Geometric Series will converge to provided that

If then the series will diverge.

∑𝒏=𝟏

∞

(𝟏𝟕 )𝒏

=¿ 𝒓=𝟏𝟕

<𝟏𝒂=𝟏𝟕 𝒄𝒐𝒏𝒗 .𝟏

𝟕+𝟏𝟕∙𝟏𝟕

+𝟏𝟕 (𝟏𝟕 )

𝟐

+𝟏𝟕 (𝟏𝟕 )

𝟑

+⋯

Section 10.2 – Infinite Series

∑𝑛=1

∞

𝑛2 lim𝑛→∞

(𝑛 )2=∞ The limit does not exist, therefore it diverges.

∑𝑛=1

∞ 𝑛+1𝑛

lim𝑛→∞

𝑛+1𝑛

=1 The limit does not equal 0, therefore it diverges.

∑𝑛=1

∞1𝑛

lim𝑛→∞

1𝑛

=0 The limit equals 0, therefore the nth – Term Test for Divergence cannot be used.

Section 10.2 – Infinite Series

Section 10.2 – Infinite Series

Telescoping Series (collapsing series)

∑𝒏=𝟏

∞

𝒂𝒏−𝒂𝒏+𝟏

A Telescoping Series will converge

∑𝒏=𝟐

∞

(𝟏𝒏− 𝟏𝒏−𝟏 )=¿𝟏

𝟐−𝟏𝟏

+𝟏𝟑−𝟏𝟐

+𝟏𝟒−𝟏𝟑

+⋯¿−𝟏