Integral & Comparison Test
Section 8.3
The Integral Test:
If fis a continuous, positive, AND decreasing function on [1,and fnanfor all
integers n≥1. Then
If 1
fxdx is convergent, then ∑
n1
anis convergent, AND
If 1
fxdx is divergent, then ∑
n1
anis divergent.
Note: We do not have to necessarily start at n1. The assumptions just need to be true
for some fixed positive integer N.
The Comparison Test:
Suppose ∑anand ∑bnare series with positive terms such that an≤bnfor all n≥1(or
any other positive integer). Then
If ∑bnis convergent, then ∑anis convergent.
If ∑anis divergent, then ∑bnis divergent.
Limit Comparison Test:
Suppose ∑anand ∑bnare series with positive terms such that
lim
n→an
bn
c
where cis some real number and c0. Then either BOTH series converge OR BOTH
diverge.