1
Convergence of Series
Last time we introduced the first important test: the integral test. The integral
test says the following:
Theorem:
Integral Test
Suppose
f
(
x
) is continuous function which is
eventually decreasing and positive, and
a
n
=
f
(
n
). Then either
∑
∞
1
a
n
∞
1
f
(
x
)
dx
both converge, or they both diverge.
This is a convenient test because it ALWAYS gives an definite outcome
(assuming that one can test convergence of
ONE
of the series. Let’s look at
some examples:
Example:
The series
∞
n
=1
1
n
p
Converges if
p >
1 and diverges if
p
≤
1. This is easy to see by applying the
integral test. It is easy to see that
f
(
x
) =
1
x
p
is decreasing for
p >
0, and is
continuous on [1
,
∞
). Computing the integral
lim
R
→∞
R
1
dx
x
p
=
lim
R
→∞
x
1

p
1

p

R
1
(1)
=
lim
r
→∞
1
1

p
(1

R
1

p
) =
0
p >
1
∞
p <
1
(2)
In the case
p
= 1 one gets
lim
R
→∞
R
1
dx
x
=
lim
R
→∞
ln(
x
)

R
1
=
lim
R
→∞
ln(
R
) =
∞
Aside:
The function
f
(
p
) =
∞
n
=1
1
n
p
is called the Riemann Zeta function. There is a conjecture (the Riemann hy
pothesis) that says that
f
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 Spring '08
 Bronski
 Math, Calculus, lim, Riemann hypothesis

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