notes2 - Infinite Sequences If you remember all of the...

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Unformatted text preview: Infinite Sequences If you remember all of the paradoxes brought up last time, the key problem was dealing with an infinite number of things. An infinite number of things should not be news to you. For example, the real line contains infinitely many points. The type of infinity were dealing with here is one that we can count 1 , 2 , 3 , 4 ,... . A infinite sequence is just a countable list of real numbers Examples of infinite sequences 1 , 1 2 , 1 3 , 1 4 ,... (1) 1 , 1 2 , 1 4 , 1 8 ,... (2) { 1 , 1 , 2 , 3 , 5 , 8 ,... } (3) Math 9C Fall 2011 (UCR) Pro-Notes September 26, 2011 1 / 1 Were going to be working a lot with infinite sequences, so we need some notation that makes things easier for us. To describe a general infinite sequence we use the notation { a 1 , a 2 , a 3 ,... } or sometimes we write { a i } i =1 or if were lazy, we just write { a i } . We also may use x i or y i or any other letter sub i Sometimes we doent want our sequence to start when i = 1, maybe we want it to start from i = 0 or i = 2 or even i = 12 whatever makes sense. In this case we would have { a i } i =0 { a i } i =2 { a i } i =12 Math 9C Fall 2011 (UCR) Pro-Notes September 26, 2011 2 / 1 How do you describe the n-th term of the sequence? Consider the first example 1 , 1 2 , 1 3 , 1 4 ,... . Here the first term has a 1 in the denominator, the second a 2, the third a 3, etc,... So to describe the n-th term we write a n = 1 n . So to describe the whole sequence we write 1 i i =1 We could have also used a different letter than i . For example, the same sequence is described by 1 j j =1 1 k k =1 1 n n =1 Math 9C Fall 2011 (UCR) Pro-Notes September 26, 2011 3 / 1 The second example...
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This note was uploaded on 02/22/2012 for the course MATH 9c taught by Professor C during the Fall '11 term at UC Riverside.

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notes2 - Infinite Sequences If you remember all of the...

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