Calculus: Nth Term Test for Divergence
Introduction
The nth-term test for divergence is a fundamental tool in calculus used to determine whether an infinite
series converges or diverges. It is often the first test applied when analyzing a series. The test states:
P
If limn→∞ an ̸= 0, then the series an diverges.
However, if limn→∞ an = 0, the test is inconclusive. The series may converge or diverge, and further
analysis is required.
Examples of Divergence
Geometric Growth
Consider the series:
2n = 20 + 21 + 22 + 23 + 24 + . . .
This series grows without bound:
1 + 2 + 4 + 8 + 16 + · · · = ∞
Thus, the series diverges.
Examples of Convergence
Geometric Decay
Consider the series:
3 3 3 3
+ + + + ...
10 100 1000 10000
1
This is a geometric series with ratio r = 10 . Since |r| < 1, the series converges:
0.3 + 0.03 + 0.003 + 0.0003 + · · · = 0.333 . . .
Logic Analogy
To understand the contrapositive nature of the test, consider the logical statement:
• Statement: If Catholic, then Christian (True).
• Converse: If Christian, then Catholic (False).
• Contrapositive: If not Christian, then not Catholic (True).
Similarly, in the nth-term test:
If a series converges, then limn→∞ an = 0.
The contrapositive is:
If limn→∞ an ̸= 0, then the series diverges.
1
Introduction
The nth-term test for divergence is a fundamental tool in calculus used to determine whether an infinite
series converges or diverges. It is often the first test applied when analyzing a series. The test states:
P
If limn→∞ an ̸= 0, then the series an diverges.
However, if limn→∞ an = 0, the test is inconclusive. The series may converge or diverge, and further
analysis is required.
Examples of Divergence
Geometric Growth
Consider the series:
2n = 20 + 21 + 22 + 23 + 24 + . . .
This series grows without bound:
1 + 2 + 4 + 8 + 16 + · · · = ∞
Thus, the series diverges.
Examples of Convergence
Geometric Decay
Consider the series:
3 3 3 3
+ + + + ...
10 100 1000 10000
1
This is a geometric series with ratio r = 10 . Since |r| < 1, the series converges:
0.3 + 0.03 + 0.003 + 0.0003 + · · · = 0.333 . . .
Logic Analogy
To understand the contrapositive nature of the test, consider the logical statement:
• Statement: If Catholic, then Christian (True).
• Converse: If Christian, then Catholic (False).
• Contrapositive: If not Christian, then not Catholic (True).
Similarly, in the nth-term test:
If a series converges, then limn→∞ an = 0.
The contrapositive is:
If limn→∞ an ̸= 0, then the series diverges.
1
