Calculus Animations,Graphics and Lecture Notes Infinite Sequences
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 An Infinite sequence is a function f(n) whose domain is an infinite subset of whole numbers.We usually write a sequence {an} . where f(n) =  anThe following 4 animations are the graphs of the sequences{(.5)n}    {n/(n+1)}    {en/n!} and {(-1)n}You'll notice that sequences are discontinuous everywhere but never the less play an important role in Mathematics as we will see when we consider Infinite Series. an = (.5)^n an = n/ (n+1) an = e^n/n! an = (-1)^n As you could tell from the animations the 1st and 3d converge to 0, the second converges to 1 and the 4th diverges.But do we mean by convergence exactly?Informally by lim an = L we mean for any ε there is a point beyond which L-  ε < an < L +  ε. for every n.Formally : given any  ε>0 there exists a number M such that L-  ε < an < L +  ε.whenever n> M Or as is usually stated  given any  ε>0 there exists a number M such that   |  an - L | <  εwhenever n > M.The following animation shows this. Animation Limit Demonstration How do we compute limits?Thee are 2 very important theorems1. Given {an} if there is a continuous differentiable  function f(x) such that f(n) = anthen lim an = lim f(x)  therefore we can use our results on differentiable functions and L'Hopital's Rule.Examples 1 and 2 fit into this category. But what about the 3d? the Gamma Function notwithstanding we need the followingThm   If  {an}  is increasing and bounded above or decreasing and bounded below it converges.The first 3 animations make this fairly obvious,However for a Fromal Proof click the link below Proof for increasing and bounced sequences Alternating SequencesAnother important theoremSuppose bn = (-1)n an where an >0.Then  if  lim an = 0 then lim bn = 0If lim an  ≠ 0  then  { bn} diverges.   Even if  lim an = L   The alternating sequence  (-1)n an diverges as the subssequence of even terms converges to L and the subssequence of odd terms converges to - L therefore the sequence diverges.The next 2 animations show the divergent case and the 3d shows convergence of an alternating sequence. (-1)^n (n/(n+1)) (-1)^n * n Converegent (-1)^n (.5)^n I would be remiss if I didn't mention the Fiboancci Sequence:1,1,2,3,5,8.........,an-1+an-2 ...........However there are entire websites devoted to this important sequence both mathematical and mystical so I'll not even attempt to begin a discussion.

The Smart Bunny-A very short story by Kurt Vonnegut Jr.