Calculus II Notes 12.1 - Calculus II-Stewart Dr. Berg...

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Calculus II- Stewart Dr. Berg Spring 2010 Page 1 12.1 12.1 Sequences The Basics Definition A sequence of real numbers is a real-valued function of natural numbers and is denoted a 1 , a 2 , , a n , { } , a n { } , or, a n { } n = 1 . Example A a) a : N R defined by a ( n ) = n 2 . i.e. {1, 4, 9, 16, 25, …} b) b : N R defined by b ( n ) = n n + 1 . i.e. {1/2, 2/3, 3/4, 4/5, …} Example B Find a formula for the general term a n of the sequence 1, 2 3 , 4 9 , 8 27 , { } . Solution : This sequence alternates sign. This can be accomplished with 1 ( ) n or 1 ( ) n + 1 . Note that the numerators are powers of 2 and the denominators are powers of 3. Hence, a n = 1 ( ) n + 1 2 3 n 1 . Example C The Fibonacci sequence f n { } is defined recursively by f 1 = 1 , f 2 = 1 , and for n > 2 , f n = f n 1 + f n 2 . Thus f n { } = 1,1,2,3,5,8,13,21, { } . Definition Let a ( n ) = a n be a sequence of real numbers and L a real number. Then lim n →∞ a n = L if for each ε > 0 there exists a number K N such that n K implies a n L < . In other words, one can make a n arbitrarily close to L by choosing n to be large enough. The point is: if a skeptic chooses some tiny distance ε you show that if one looks far enough out in the sequence (past K ) all the elements of the sequence are within that tiny distance of the real number L .
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This note was uploaded on 04/04/2011 for the course MATH 408 L taught by Professor Zheng during the Spring '10 term at University of Texas at Austin.

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Calculus II Notes 12.1 - Calculus II-Stewart Dr. Berg...

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