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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) = an
The 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 | < ε
whenever n > M.
The following animation shows this.
Animation Limit Demonstration
How do we compute limits?
Thee are 2 very
important theorems
1. Given {an} if there is a continuous differentiable function f(x) such that
f(n) = an
then 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 following
Thm 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 Sequences
Another important theorem
Suppose bn = (-1)n an where an >0.
Then if lim an
= 0 then lim bn = 0
If 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
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