AP Calculus BC · Lesson 1 of 4
Sequences and series
Distinguish a list of terms from the sum of those terms, and justify convergence with a test whose conditions you have checked.
Sequences versus series
A sequence is an ordered list, such as 1, 1/2, 1/3, 1/4. A series adds the terms of a sequence. The partial sums form a new sequence, and the series converges when those partial sums approach a finite number.
If the terms of a series do not approach zero, the series diverges. The converse is false: terms can approach zero while the series still diverges, as with the harmonic series. “The terms get small” is not a convergence proof.
Geometric series and error
An infinite geometric series a + ar + ar² + … converges when |r| < 1, and its sum is a/(1 − r). You must state the ratio and confirm that absolute value before using the formula. For 4 − 2 + 1 − 1/2 + …, a = 4 and r = −1/2, so the sum is 4/(1 − (−1/2)) = 8/3.
A requested approximation is not settled by convergence alone. A remainder estimate or error bound says how far a partial sum can be from the infinite sum. Name the test or remainder result you use.
Worked example
Does the geometric series with first term 5 and common ratio 2 converge?
- The ratio is r = 2.
- |r| = 2, which is not less than 1.
- The series diverges. The formula a/(1 − r) does not apply.
Why this works. Check |r| < 1 before summing an infinite geometric series.
Check your understanding
- Say whether the object is a sequence, a series, or a sequence of partial sums.
- Use “terms go to zero” only as a necessary condition for convergence.
- Pair a convergence conclusion with an error bound when an approximation is requested.