Section 11.1 Sequences

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1 Math 152 c Lynch 1 of 8 Section 11.1 Sequences A sequence is a list of numbers written in a definite order: a 1, a 2, a 3,..., a n,... Notation. The sequence {a 1, a 2, a 3,...} can also be written {a n }, {a n } n=1, or a n = f(n). Examples of Sequences. (with different notations shown) 1. { 1, 1 2, 1 3, 1 4,..., 1 n,...}, a n = 1 n, { 1 n} n=1 { } , 4 25, 5 125, 6 625,..., ( 1)n (n+2) 5,..., a n n = ( 1)n (n+2) 5 n, { } ( 1) n (n+2) 5 n 3. { 0, 1, 2, 3,..., n 3,... }, a n = { } n 3, n 3, n 3 n=3 { 4. 1, 3 2, 1 2, 0,..., cos ( ) } πn 6,..., a n = cos ( ) { ( πn 6, n 0, cos πn 6 )} n=0 Example 1. Find a formula for the general term a n of the following sequences assuming the pattern of the first few terms continues. { (a) 7, 14 3, 28 9, 56 27, , 224 } 247,... (b) { 10, 2, 2 5, 2 25, 2 } 125, (c) { 5 4, 7 16, 9 64, 11 } 256, ,...

2 Math 152 c Lynch Section of 8 Examples of Sequences without a simple defining equation: 1. Let a n be the n th prime number. Then {a n } is a well-defined sequence with first terms {2, 3, 5, 7, 11, 13, 17, 19,...} 2. The Fibonacci Sequence {f n } is defined recursively. f 1 = 1, f 2 = 1, and f n = f n 1 + f n 2, for n 3 Then the sequence is {1, 1, 2, 3, 5, 8, 13, 21,...} There is a complicated formula for the Fibonacci Sequence: f(n) = φn (1 φ) n where φ = Graphs. A sequence can be graphed by plotting its terms on a number line or in a 2-dimensional plane. Example 2. a n = n n+1 Real Definition. If {a n } is a sequence, then lim a n = L n means that for every ɛ > 0 there is a corresponding integer N such that if n > N, then a n L < ɛ. Intuitive Definition. A sequence {a n } has the limit L and we write lim a n = L or a n L as n n if we can make the terms a n as close to L as we like by taking n to be sufficiently large. If lim n exists, we say that the sequence converges (or is convergent). Otherwise, we say the sequence diverges (or is divergent).

3 Math 152 c Lynch Section of 8 What is the difference between lim a n = L and lim f(x) = L? n x Theorem. If lim x f(x) = L and f(n) = a n, when n is an integer, then lim a n = L n Example 3. Determine if {ne n } converges or diverges. Infinite Limits. If a n becomes arbitrarily large as n becomes large, then we use the notation lim n a n =. But the sequence is still considered divergent. Limit Rules: If {a n } and {b n } are convergent sequences and c is a constant, then the following are true. 1. lim n (a n + b n ) = lim n a n + lim n b n 2. lim n ca n = c lim n a n 3. lim n (a nb n ) = lim n a n lim n b n a n 4. lim = lim n a n, if lim n b n lim n b b n 0 n n 5. lim n (a n) p = [ lim n a n ] p if p > 0 and a n 0

4 Math 152 c Lynch Section of 8 Squeeze Theorem for Sequences. If a n b n c n for n n o and lim a n = lim c n = L then lim b n = L n n n Example 4. Determine whether the sequences converge or diverge. If the sequence converges, determine what it converges to. n (a) a n = 1 + n (b) a n = ln n n Example 5. Prove that if lim n a n = 0, then lim n a n = 0

5 Math 152 c Lynch Section of 8 Definition. An alternating sequence is a sequence that alternates between positive and negative terms. Theorem. An alternating sequence only converges if it converges to 0. Therefore, an alternating sequence converges if and only if the absolute value of the terms converges to 0. Example 6. Determine whether the following sequences converge or diverge. If the sequence converges, determine what it converges to. (a) a n = ( 1)n+1 5n n 2 (b) a n = ( 1) n ne n (c) a n = ( 1)n n! n n

6 Math 152 c Lynch Section of 8 Example 7. For what values of r is the sequence {r n } convergent? Theorem. The sequence {r n } is convergent if 1 < r 1 and divergent for all other values of r. 0 if 1 < r < 1 lim n rn = 1 if r = 1 Definition. A sequence {a n } is called increasing if a n < a n+1 for all n 1, that is a 1 < a 2 < a 2 < < a n. It is called decreasing if a n > a n+1 for all n > 1, that is a 1 > a 2 > a 3 > > a n >. A sequence is monotonic if it is either increasing or decreasing. Example 8. Determine if the sequence is increasing, decreasing, or not monotonic. (a) a n = 3 + ( 1)n n (b) a n = n n 2 +1

7 Math 152 c Lynch Section of 8 Definition. 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 a n M for all n 1. If it is bounded above and below, then {a n } is a bounded sequence. Monotonic Sequence Theorem: Every bounded, monotonic sequence is convergent. Example 9. Determine if the sequence a n = 4 2n + 7 is convergent or divergent. Application to Sequences: For recursive sequences that are monotonic and bounded, we can use mathematical induction to prove the sequence is monotonic and bounded, and therefore convergent by the Monotonic Sequence Theorem. Principle of Mathematical Induction: Let S n be a statement about the positive integer n. If 1. S 1 is true. 2. If S k is true, then S k+1 is true. Then S n is true for all positive integers n.

8 Math 152 c Lynch Section of 8 Increasing: Suppose the following are true. 1. a 1 < a 2 2. If a k < a k+1, then a k+1 < a k+2. Then a n is an increasing sequence by mathematical induction. (Similarly, if each < is replaced by >, then a n is a bounded, decreasing sequence.) Bounded: Let A and B be constants, and suppose the following are true. 1. A a 1 B 2. If A a k B, then A a k+1 B. Then a n is a bounded sequence by mathematical induction. Example 10. Determine if the following sequence is convergent or divergent. 1 a 1 = 1 and a n+1 = 6 3a n