L22 - 11.1 - Sequences - key (1) - CALCULUS 2 NAME LAB 22...

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CALCULUS 2NAME: ________________________________LAB 22 11.1 SEQUENCESLab Time: ____________ Date: __________Basic Notions:* A sequence is a function whose domain is a set of integers. f(n)an,n1,2,3,....* A sequence {an}n1converges to L (finite), if limnanL.* A sequence {an}n1has at most a single limit L (finite). If the terms naare not getting closer to a single finite value, we say that the sequence diverges.* Ifx[1,),thenlimxf(x)Llimnf(n)limnanL.* A sequence is increasingif its terms are getting larger:1nnaa* A sequence is decreasingif its terms are getting smaller:1nnaaThere are three simple ways to check whether the given sequence {na}is increasing or decreasing:1. Subtraction Method: If 01nnaa, then the sequence is increasingIf 01nnaa, then the sequence is decreasing2. Division Method: If 1/1nnaa, then the sequence is increasingIf 1/1nnaa, then the sequence is decreasing3. Derivative Method: Replace the n’s in the series with x’s to create a similar function )(xfand look at)(xfIf 0)('xffor 1x, then the sequence is increasingIf 0)('xffor 1x, then the sequence is decreasing.Write out the first four terms of the sequence, try to find its limit, and determine whether the sequence converges or diverges.