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Summary Sequences | Calculus II Notes
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Includes Definitions, notations, and examples. Explains convergence to a limit, monotone sequence theorem, and bounding terms.

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  • July 14, 2023
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  • 2022/2023
  • Summary

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3.1 Sequences
definition:
a sequence is an infinite listindexed by IN 51,2, 3....3 Covany [a,a +1, a +2. I often
=

...




starting at0



Notation:

C., C2, C3...
ECn3,.. or [Cn]

eg. Cn in"
for nyl explicitformula
=




Emma
(n nCn-1, for ne formula
=


recursive

(, 1, (n
=

2.1,
=



2z 3)2
=
3.2.1
=




(n n(n-1)....
=
3.2



eg. An=(-1)" or An=Cos(in) for n =0

1, -1,1,-.




a sequence [an3 converges to limit ((himAn=1) ifwe can make an as close as we can

to L by requiring to be sufficiently large.
dany diverges otherwise ( An dne)

eg. 0Un=cos (in)
ing (OS (in) dne

② lz 0
=
Sins converges to 0



if f is defined on R (or any continuous domain), **f(x) 2
=


and if an=f(n), then
liAn L
=




eg. O Re
x
0
-


=
f(x) e
=




② arctan(n) Fz =




③ In (In) = -

-(dne)
Sen(/n) 3 diverges

all limit rules apply provided all limits existand the limitofthe denominator of a quotient to

ifan=br? In for all n and han=(n=2 then limba c
=




ifg is continuous and Ran=2 then ig(an) g() =




eg. 02 Arctan (In)=arctan (0) 0
=




② An=
I
-

In an in en =0, 3 converges 0
to

!
~Ov

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