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Unformatted text preview: Sequences CheatSheet 1 Intro This cheatsheet contains everything you should know about real sequences. It’s not meant to be exhaustive, but it contains more material than the textbook. 2 Definitions and properties In this first section we define the notion of sequence of real numbers and the notion of subsequence . Definition 2.1. A sequence of real numbers a n is a function whose domain is the natural numbers and range is contained in the real numbers: N a · / R n / a n Definition 2.2. A sequence a n is called increasing if a i < a i +1 for all i ∈ N . It’s called decreasing if a i > a i +1 for all i ∈ N . It’s called monotonic if it’s either increasing or decreasing. Definition 2.3. Let p be an increasing sequence of natural numbers (it’s range is contained in N ). Let a n be a sequence, we call a subsequence of a n the composition a p n : N p · / N a · / R k / p k / a p k Example 2.4 . For example let p k = 2 k and a n is any sequence; the composition a p k = a 2 k is the subsequence of all the even members of a n . If p k = 2 k +1, the composition a 2 k +1 is the subsequence of all the odd members. Definition 2.5. A sequence a n is bounded from above if there is a real number L such that a n ≤ L for all n ∈ N . It is bounded from below if there is a real L such that a n ≥ L for all n ∈ N . It’s bounded if it is bounded from above and below. 3 Convergence A real sequence a n is convergent to a real number L if you can make a n arbitrarily close to L by taking n large enough. Definition 3.1. We say that a n converges to L and we denote it with lim n → + ∞ a n = L if ∀ ∈ R , > 0 there exists n ( ) ∈ N such that ∀ n > n ( ) , | a n- L | < . 1 In this definition measures how close we are to L and n ( ) tells us how large “large enough” is. Example 3.2 . Let a n be the sequence 1 n p where p is a positive real number. Pick any number > 0. The condition: 1 n p < is fulfilled if: n >- 1 p The number- 1 p is positive but not natural, so we put n ( ) = l- 1 p m This choice of n ( ) proves that a n converges to zero. The writing d x e denotes the ceiling function which maps x to the smallest integer which is greater than or equal to x . A real sequence a n is divergent if you can make a n arbitrarily large (positive or negative) by taking n large enough. Definition 3.3. We say that a n is divergent and denote it with: lim n → + ∞ a n = + ∞ if ∀ C ∈ R there exists n ( C ) ∈ N such that ∀ n > n ( C ) , a n > C . Example 3.4 . Let a n be the sequence ln( n ). Let C be any positive real number. The condition ln( n ) > C is satisfied if we pick n > e C because the exponential is an increasing function. We prove that a n is divergent by choosing n ( C ) = e C ....
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This note was uploaded on 11/03/2011 for the course MATH 1090 taught by Professor Greenwood during the Spring '08 term at MIT.
- Spring '08