Ch. 10 – Infinite Series 9.1 – Sequences

Sequences Infinite sequence = a function whose domain is the set of positive integers a 1, a 2, …, a n are the terms of the sequence Ex: Find the first four terms and the 33 rd term of a n = 4 + 2n a 1 = 4 + 2(1) = 6 a 2 = 8, a 3 = 10, a 4 = 12 a 33 = 4 + 2(33) = 70 You can enter this function into your calculator and use the TABLE to check your answers.

Recursive Function = a sequence that uses previous terms as inputs Ex: If a 0 = 1, a 1 = 1, and a n = a n-2 + a n-1 for n ≥ 2, find a 5. – a 2 = = 2 – a 3 = = 3 – a 4 = = 5 – a 5 = = 8 Factorial Notation: n! = 1(2)(3)(4)…(n-1)(n) It’s the Fibonacci sequence!

A sequence is arithmetic if the difference between consecutive terms is constant ( ex: 3, 7, 11, 15, …) The explicit rule for arithmetic sequences is a n = d(n – 1) + a 1 = dn + a 0 d = the common difference between successive terms The recursive rule for arithmetic sequences is a n = a n – 1 + d Ex: Write the explicit and recursive rules for the sequence: -4, 3, 10, 17, … EXPLICIT RECURSIVE

A sequence is geometric if consecutive numbers always have the same common ratio (r). Ex: has a common ratio of -2/3 The explicit rule for geometric sequences is a n = a 1 (r n-1 ) = a 0 (r n ) r = the common ratio between successive terms The recursive rule for geometric sequences is a n = r (a n – 1 )

Ex: If the 4 th term of a geometric sequence is 48 and the 10 th term is 3/4, find the 14 th term. Think about the relationship between the 10 th and 4 th terms!

Summations (or Series) The sum of the first n terms of a sequence is represented by: where k is the index (starting value) and n is the limit of the summation Ex: Find. Add a 1 through a 4 ! This value is called a partial sum!

Properties of Sums: If c is a constant, then… The sum of the first n terms of a finite sequence is called a partial sum or a finite series The sum of all terms in an infinite sequence is called an infinite series

Convergence If the terms of a sequence approach a finite value L as n approaches ∞, then we say the sequence converges on L. – Convergence means – A sequence that does not converge is said to diverge. Ex: Determine whether the sequence converges or diverges. If it converges, find its limit. – Converges to 3/2! – Function oscillates between -1 and 1, so… Diverges! By L’Hopital’s Rule…

The sum of an infinite geometric sequence is: – If |r| ≥ 1, the series does not have a sum. Ex: Find the sum: – Use the infinite sum formula! – To find a 1, evaluate for k = 1 a 1 = 4(0.6) 1-1 = 4(0.6) 0 = 4

Convergence Some sequences that don’t converge: – a n = n(1, 2, 3, 4, …) – Arithmetic sequences (2, 0, -2, -4, …) – Geometric sequences with r > 1 (1, 1.1, 1.21, 1.331, …) – Rational functions with the degree in the numerator greater than the degree in the denominator Geometric series will converge on their infinite sum, if one exists! Ex: Determine whether the series converges or diverges. If it converges, find its limit. – Diverges because 2.4 > 1 – Converges to 8/3!

Sandwich Theorem for Sequences Given 3 sequences a n, b n, and c n, iffor a common domain of n and, then. Ex: Determine whether the sequence converges or diverges. If it converges, find its limit. – Consider c n = 1/n, which we know converges to zero. Since cosn will always be between -1 and 1… – Therefore, by Sandwich Thm., a n converges to zero.