The ratio test
The next test we’ll learn about is called the ratio test. It’s really the best
test out there. And we’ll see later that it’s also a very important test when
we start to study power series.
The idea behind the ratio test is this: In a geometric series the ratio of two
consecutive terms was always
r
, by this I mean if
a
k
=
r
k
, then
a
k
+1
a
k
=
r
k
+1
r
k
=
r
for every
k
. And we know that a geometric series converges if

r

<
1 and
diverges if

r
 ≥
1. The ratio test says: if you don’t have a geometric
series, but you do have that “in the limit” the ratio of two consecutive
terms is some number
ρ
(funny Greek letter for
r
called rho), our series
will behave similar to a geometric series.
Math 9C Summer 2011 (UCR)
ProNotes
October 14, 2011
1 / 1
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View Full DocumentTheorem (The ratio test)
Let
∑
∞
k
=1
a
k
be an inﬁnite series. Set
ρ
= lim
k
→∞
±
±
±
±
a
k
+1
a
k
±
±
±
±
Then
I
If
ρ <
1
, the series converges (it even converges absolutely!).
I
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 Fall '11
 c
 Calculus, Power Series, Mathematical Series, lim

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