Section 9.1 – Sequences

Sequence A sequence { 𝑎 𝑛 } is a list of numbers written in an explicit order. 𝑎 𝑛 ={ 𝑎 1 , 𝑎 2 , 𝑎 3 , …, 𝑎 𝑛 ,…} First Term Second Term nth Term Generally, we will concentrate on infinite sequences, that is, sequences with domains that are infinite subsets of the positive integers.

𝑎 1 =2 𝑎 𝑛 =2∙ 𝑎 𝑛−1 Recursive Formula A formula that requires the previous term(s) in order to find the value of the next term. Example: Find a Recursive Formula for the sequence below. 2, 4, 8, 16, … 𝑎 1 =2 𝑎 𝑛 =2∙ 𝑎 𝑛−1

Explicit Formula A formula that requires the number of the term in order to find the value of the next term. Example: Find an Explicit Formula for the sequence below. 2, 4, 8, 16, … The Explicit Formula is also known as the General or nth Term equation.

Arithmetic Sequences A sequence which has a constant difference between terms. The rule is linear. Example: 1, 4, 7, 10, 13,… (generator is +3) n a(n) 1 2 4 3 7 10 5 13 +3 Discrete +3 +3 Explicit Formula Recursive Formula +3

Sequences typically start with n=1 White Board Challenge Write an equation for the nth term of the sequence: Then find the n=0 term. n=0 n=1 n=2 n=3 n=4 36, 32, 28, 24, … 40, – 4 First find the generator Sequences typically start with n=1 a(0) is not in the sequence! Do not include it in tables or graphs!

Geometric Sequences A sequence which has a constant ratio between terms. The rule is exponential. Example: 4, 8, 16, 32, 64, … (generator is x2) n t(n) 1 4 2 8 3 16 32 5 64 x2 Discrete x2 x2 0 1 2 3 4 5 6 Explicit Formula Recursive Formula x2

Sequences typically start with n=1 White Board Challenge Write an equation for the nth term of the sequence: Then find the n=0 term. n=0 n=1 n=2 n=3 n=4 3, 15, 75, 375, … x5 First find the generator Sequences typically start with n=1 a(0) is not in the sequence! Do not include it in tables or graphs!

New Sequences The previous sequences were the only ones taught in Algebra 2. But, it is possible for a sequence to be neither arithmetic nor geometric. Example: Find a formula for the general term of the sequence below 3 5 , − 4 25 , 5 125 , − 6 625 , 7 3125 , … n=1 n=2 n=3 n=4 n=5

White Board Challenge 1, 1 3 , 1 5 , 1 7 , 1 9 , … Example: Find a formula for the general term of the sequence below 1, 1 3 , 1 5 , 1 7 , 1 9 , … n=1 n=2 n=3 n=4 n=5

Monotonic Sequence A sequence is monotonic if it is either increasing (if 𝑎 𝑛 < 𝑎 𝑛+1 for all 𝑛≥1) or decreasing (if 𝑎 𝑛 > 𝑎 𝑛+1 for all 𝑛≥1). Example 1: Find the first 4 terms of 𝑎 𝑛 = 𝑛 𝑛+1 to see how the sequence is monotonic.

Example 2 Prove the sequence 𝑎 𝑛 = 3 𝑛+5 is decreasing. If the sequence is decreasing, 𝒂 𝒏 > 𝒂 𝒏+𝟏 for all 𝒏. IF: THEN: Since the denominator is smaller: OR Therefore, 𝒂 𝒏 is decreasing.

Bounded Sequence A sequence { 𝑎 𝑛 } is bounded above if there is a number 𝑀 such that 𝑎 𝑛 ≤𝑀 for all 𝑛≥1 A sequence { 𝑎 𝑛 } is bounded below if there is a number 𝑚 such that 𝑚≤ 𝑎 𝑛 for all 𝑛≥1 If it is bounded above and below, then { 𝑎 𝑛 } is a bounded sequence.

Therefore, 𝒂 𝒏 is bounded below. Therefore, 𝒂 𝒏 is bounded. Example Determine if the sequences below bounded below, bounded above, or bounded. 𝑎 𝑛 =𝑛 𝑎 𝑛 = 𝑛 𝑛+1 Therefore, 𝒂 𝒏 is bounded below. Since 𝒏=𝟏, 𝟐,𝟑,… : 𝒂 𝒏 ≥1 Since lim 𝒏→∞ 𝒏 =∞ : The sequence is not bounded above. Since 𝒏=𝟏, 𝟐,𝟑,… : 𝒂 𝒏 >0 Therefore, 𝒂 𝒏 is bounded. Since lim 𝒏→∞ 𝒏 𝒏+𝟏 =𝟏 : 𝒂 𝒏 <1.

Limit of a Sequence A sequence { 𝑎 𝑛 } has the limit 𝐿 and we write: lim 𝑛→∞ 𝑎 𝑛 =𝐿 or 𝑎 𝑛 →𝐿 as 𝑛→∞ if we can make the terms 𝑎 𝑛 as close to 𝐿 as we like by taking 𝑛 sufficiently large. If lim 𝑛→∞ 𝑎 𝑛 exists, we say the sequence converges (or is convergent). Otherwise, we say the sequence diverges (or is divergent).

Reminder: Properties of Limits Let b and c be real numbers, let n be a positive integer, and let f and g be functions with the following limits: Constant Function Limit of x Limit of a Power of x Scalar Multiple

Reminder: Properties of Limits Let b and c be real numbers, let n be a positive integer, and let f and g be functions with the following limits: Sum/Difference Product Quotient Power

Example 𝑎 𝑛 = 𝑛 𝑛+1 𝑎 𝑛 = 𝑛 10+𝑛 𝑎 𝑛 = ln 𝑛 𝑛 Determine if the sequences below converge or diverge. If the sequence converges, find its limit. 𝑎 𝑛 = 𝑛 𝑛+1 𝑎 𝑛 = 𝑛 10+𝑛 𝑎 𝑛 = ln 𝑛 𝑛 Converges to 1 Diverges Converges to 0

White Board Challenge Determine whether the sequence converges or diverges. If it converges, find its limit. 2 2 −2 2 2 , 3 2 −2 3 2 , 4 2 −2 4 2 , 5 2 −2 5 2 , … 𝑎 𝑛 = (𝑛+1) 2 −2 (𝑛+1) 2 Converges to

Absolute Value Theorem It is not always possible to easily find the limit of a sequence. Consider: −1, 1 2 , − 1 3 , 1 4 ,…, −1 𝑛 𝑛 ,… The Absolute Value Theorem states: If lim 𝑛→∞ 𝑎 𝑛 =0, then lim 𝑛→∞ 𝑎 𝑛 =0.

Example Determine if the sequences below converge or diverge. 𝑎 𝑛 = −1 𝑛 𝑛 𝑎 𝑛 = −1 𝑛 Because of the Absolute Value Theorem, Converges to 0 Since the limit does not equal 0, we can not apply the Absolute Value Theorem. It does not mean it diverges. Another test is needed. The sequence diverges since it does not have a limit: -1,1,-1,1,-1,…

Theorem: Bounded, Monotonic Sequences Every bounded, monotonic sequence is convergent. Example: Investigate the sequence below. 𝑎 1 =2 𝑎 𝑛+1 = 1 2 ( 𝑎 𝑛 +6) Since the sequence appears to be monotonic and bounded, it appears to converge to 6. The sequence appears to be monotonic: It is increasing. The sequence appears to be bounded: 2≤ 𝑎 𝑛 ≤6 The limit of the sequence appears to be 6.