Calculus: Limit Comparison Test
Definition
The Limit Comparison
P Test
P is used to determine convergence of series with positive terms.
Given two series an and bn with an , bn > 0, compute:
an
L = lim
n→∞ bn
Convergence Criteria
• If 0 < L < ∞, then either both series converge or both diverge.
• If L = 0 and
P P
bn converges, then an also converges.
• If L = ∞ and
P P
bn diverges, then an also diverges.
Examples
Example 1
3n2 − 4n + 5 1
an = , bn =
n3 n
an 3n2 − 4n + 5
lim = lim =3
n→∞ bn n→∞ n2
P1 P
Since L = 3, both series behave the same. n
diverges, so an diverges.
Example 2
n2 − 10 1
an = , bn =
n3 + n2 n
an n2 − 10
lim = lim 2 =1
n→∞ bn n→∞ n + n
P1 P
Since L = 1, both series behave the same. n
diverges, so an diverges.
1
Definition
The Limit Comparison
P Test
P is used to determine convergence of series with positive terms.
Given two series an and bn with an , bn > 0, compute:
an
L = lim
n→∞ bn
Convergence Criteria
• If 0 < L < ∞, then either both series converge or both diverge.
• If L = 0 and
P P
bn converges, then an also converges.
• If L = ∞ and
P P
bn diverges, then an also diverges.
Examples
Example 1
3n2 − 4n + 5 1
an = , bn =
n3 n
an 3n2 − 4n + 5
lim = lim =3
n→∞ bn n→∞ n2
P1 P
Since L = 3, both series behave the same. n
diverges, so an diverges.
Example 2
n2 − 10 1
an = , bn =
n3 + n2 n
an n2 − 10
lim = lim 2 =1
n→∞ bn n→∞ n + n
P1 P
Since L = 1, both series behave the same. n
diverges, so an diverges.
1
