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Comparison
Geometric
test
Earn
:
If
convery
2 must
e
x
-
Irk)
dx
-ex? z
Since Idiverges
P=1,
n
an
Gan
the
integrals
the
I
=
00
diverges
converges
Series
Diverges
fur=
FO
00
-
Integral
an ->
diverge
=an
If
Ebn
10f(x)dX
br
=
converges antcon.
Ifan-Div,
bn-Div.
-
Patio
Limit
Comparison Test
postisinto
"you
Lon.
Alternating
for
series we
1. Pass
LiM
alternating signs
too
Series
denom.
when
yet smaller
total frac Gets
Divergent
-
DNe
decreasing
->
positive, continous,
finite
=
2
val
00 or
I
if
from
00
n
·
converges
Series
-
DNE Diverges
test
too late
t I
root
test
is
test
result
if
Converges
(1, Series
result I/90, Series Diverges
ifresult=1, inconclusive
if
Or Dir.
Series
series
bigger
f(n)
mustbe
I
Direct
comparison test
infinite
Er when
Gettable
for Aaylur
+
100
if
of
ra
som
or
it may
p
for >1
Geometric
Partial=L- Converges
Lim
Diverges
ice
Sum
it... an
+ -!
-
* an
for
Geometric Series
series
nto
divergence test
nth term
him
by
bythe comparison
also
original
diverges
series
Telescoping
=
testfor improper
converges
Diverges
converges
an=arn-
PCI
ex
series
then the Ser:es
*PD
x2
cx=
P-test,
p
then the
P series
Example
ex
9
Importantfacts
P-test=
IfIr12)
I
converged
If
Series
= DiV
isconv
test
and
if
C1, converges
if
1 ordo, Diverges
if
=
Dir. testLim
n > 00
1,
in conclusive
-
has to 0
=
2. ABS valve test
if lanl -> Con.
then an -> Con
3
absolutely
convergent
Conditionally
S convergent
I's??Dir
If
and
lans- Div.
an -> Con.
?
an+1 an
an
where
an
-
bu
Pe
corv
just
becomes
>0?
it
Dn
masing;
i)
atteternites
Absolute
If
land
Convergence
converges
then
Converges
On
Improper Integrall
absolutely
(0f(x)dx 450f(x)dX
I
Sun
=
/I
CONV.
test
diV.
cor. Your
[und.
quTA3
!) (!)
improper integrals Conv
An improper
ithim.
integral
the p-integral for
ps 1:
cor.
100
Exists
XPis
or
called
as O
a
3. So ti
Div.
80Div. 3 I Conv
P 1:90?J. Both Diverge
p21:
=
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