MATH231 Lecture Notes 2 - 1 Sequences and Series Definition...

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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( 4 ) 0 , 2 2 , 1 , 2 2 , 0 , - 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 > 0 there exists an N such that | a n - L | < for all n > 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 > N implies | a n - 0 | < . I can choose N = - 1 / 2 Of course one basically NEVER evaluates a limit using the , δ 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 = lim x →∞ 4 6 = 2 3 1
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BY the above theorem lim x →∞ f ( x
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