WOLFRAM|DEMONSTRATIONS PROJECT

The Integral Test and Error Estimation Using p-Series

​
p
⟵ move the rectangles ⟶
show the error in theestimate of the series with
s
10
​when p>1
The area of the colored rectangles represents the sum of the series
∞
∑
n=1
a
n
. With the rectangles in the starting (left) position, you can see that
∞
∑
n=2
a
n
≤
∞
∫
1
f(x)dx
. Slide the rectangles to the right to see that
∞
∫
1
f(x)dx≤
∞
∑
n=1
a
n
. Thus the series and the integral converge or diverge together. For a convergent
p
-series (or any convergent series satisfying the criteria of the integral test), the inequality
s
N
+
∞
∫
N+1
f(x)dx≤
∞
∑
n=1
a
n
≤
s
N
+
∞
∫
N
f(x)dx
holds for all
N
, where
s
N
is the
th
N
partial sum. An option in this Demonstration lets you see this fact using
N=10
. The gray rectangles represent
s
10
, and the colored rectangles represent the "tail",
T
10
=
∞
∑
n=11
a
n
.