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Unformatted text preview: 1 Sequences and Series Definition: A sequence is a function whose domain is the set of integers: in other words for each integer we have a real number. Some examples: Example: a n = 1 n 2 or 1 , 1 4 , 1 9 , 1 16 . . . Example: a n = sin( n 4 ) 0 , 2 2 , 1 , 2 2 , ,- 2 2 ,- 1 , . . . Example: a n = 2 +- 1 n 1 , 5 2 , 5 3 , 9 4 , 9 5 . . . Definition: Convergence a n L if for every &gt; 0 there exists an N such that | a n- L | &lt; for all n &gt; N In other words, the game is this: In order for a sequence to converge the following has to hold: You give me a tolerance ( ) I have to find a number L (the limit) such that EVERY element after some point is within the tolerance ( ) of the limit ( L ). If I can do this for EVERY tolerance then the sequence is convergent. Examples: Examples 1 and 3 above are convergent. Example 2 is not convergent. Proof a n = 1- 1 n 2 . I claim that the limit is 1. In other words for every positive I can find N such that n &gt; N implies | a n- | &lt; . I can choose N =- 1 / 2 Of course one basically NEVER evaluates a limit using the , definition. The following theorem is the one which is most useful: Theorem: Suppose that lim x f ( x ) = L . Then lim n f ( n ) = L Example: Evaluate lim n 2 n 2 + 1 3 n 2 + n + 18 lim x 2 x 2 + 1 3 x 2 + x + 18 = lim x 4 x 6 x + 1...
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