Calculus: Ratio Test
Definition
The
P Ratio Test is used to determine the convergence of an infinite series. Given a series
an , compute:
an+1
L = lim
n→∞ an
Convergence Criteria
• If L < 1, the series converges absolutely.
• If L > 1 or L = ∞, the series diverges.
• If L = 1, the test is inconclusive.
Examples
Example 1: Factorial Series
1
an =
n!
an+1 1 1
= · n! =
an (n + 1)! n+1
1
lim =0<1
n→∞ n + 1
Therefore, the series converges absolutely.
Example 2: Geometric Series
n
1
an =
2
an+1 (1/2)n+1 1
= n
=
an (1/2) 2
1
Definition
The
P Ratio Test is used to determine the convergence of an infinite series. Given a series
an , compute:
an+1
L = lim
n→∞ an
Convergence Criteria
• If L < 1, the series converges absolutely.
• If L > 1 or L = ∞, the series diverges.
• If L = 1, the test is inconclusive.
Examples
Example 1: Factorial Series
1
an =
n!
an+1 1 1
= · n! =
an (n + 1)! n+1
1
lim =0<1
n→∞ n + 1
Therefore, the series converges absolutely.
Example 2: Geometric Series
n
1
an =
2
an+1 (1/2)n+1 1
= n
=
an (1/2) 2
1
