Calculus: Alternating Series
Definition of Alternating Series
An alternating series is a series whose terms alternate in sign, typically of the form:
∞
X
(−1)n+1 an
n=1
where an > 0.
Examples:
1 1 1
1− + − + ···
2 3 4
1 − 3 + 5 − 7 + ···
Alternating Series Test (Leibniz Criterion)
(−1)n+1 an converges if:
P
The series
1. an ≥ 0.
2. an+1 ≤ an (terms are decreasing).
3. limn→∞ an = 0.
If these conditions are met, the series converges. If limn→∞ an ̸= 0, the series diverges.
Error Estimation in Alternating Series
For an alternating series that converges, the error in approximating the sum by the first N
terms is less than the magnitude of the first omitted term:
|RN | ≤ aN +1
This means if you stop after 5 terms, the error is bounded by the size of the 6th term.
1
Definition of Alternating Series
An alternating series is a series whose terms alternate in sign, typically of the form:
∞
X
(−1)n+1 an
n=1
where an > 0.
Examples:
1 1 1
1− + − + ···
2 3 4
1 − 3 + 5 − 7 + ···
Alternating Series Test (Leibniz Criterion)
(−1)n+1 an converges if:
P
The series
1. an ≥ 0.
2. an+1 ≤ an (terms are decreasing).
3. limn→∞ an = 0.
If these conditions are met, the series converges. If limn→∞ an ̸= 0, the series diverges.
Error Estimation in Alternating Series
For an alternating series that converges, the error in approximating the sum by the first N
terms is less than the magnitude of the first omitted term:
|RN | ≤ aN +1
This means if you stop after 5 terms, the error is bounded by the size of the 6th term.
1
