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