Ratio and Root Tests
John E. Gilbert, Heather Van Ligten, and Benni Goetz
Properties of geometric series enable us to test whether other series converge or diverge  even if
they’re not geometric series!
Root Test:
an infinite series
n
a
n
•
Converges Absolutely
when
lim
n
→∞

a
n

1
/n
<
1
,
•
Diverges
when
lim
n
→∞

a
n

1
/n
>
1
,
•
if
lim
n
→∞

a
n

1
/n
=
1
, the test is inconclusive, it tells us
nothing.
Example 1:
does the series
∞
n
= 1
(

1)
n

1
n
2
n
+ 1
n
converge absolutely
?
Solution:
the Root Test works well here
because of the
n
th
power exponent in
a
n
= (

1)
n

1
n
2
n
+ 1
n
.
For then

a
n

1
/n
=
n
2
n
+ 1
n
1
/n
=
n
2
n
+ 1
→
1
2
as
n
→ ∞
. So the series converges
absolutely by the Root Test.
What’s the connection with geometric series? Well,
lim
n
→ ∞

a
n

1
/n
=
L
=
⇒

a
n

≈
L
n
for all large
n,
so
n

a
n

≈
n
L
n
.
But we know that the geometric series
n
L
n
converges when
L <
1
and diverges when
L >
1
. With
more care, this establishes the Root test.
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There’s another test similar to the Root Test, but one which often works well. It’s connection with
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 Spring '07
 Sadler
 Geometric Series, Multivariable Calculus, Mathematical Series

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