sequences definition - a function whose domain is the set of all positive integers. finite sequence...
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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
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 10.2 – 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.
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