A sequence is an ordered list of numbers, and convergence asks whether its individual terms get closer and closer to one fixed value. A series is the sum of the terms of a sequence, and convergence asks whether the running totals approach a finite number. This distinction matters because a sequence can settle down while the accumulated sum can still grow without bound.
In calculus, understanding this difference is essential for power series, approximations, and improper integrals.
For a sequence {a_n}, we study the limit lim n→∞ a_n. For a series Σ a_n, we study the limit of the partial sums s_n = a_1 + a_2 + ... + a_n. A necessary condition for a series to converge is that its terms must approach zero, because nonzero terms keep adding a lasting amount to the total.
However, a_n → 0 by itself is not enough to guarantee that Σ a_n converges, as shown by the harmonic series.
Key Facts
- Sequence convergence: a_n converges to L if lim n→∞ a_n = L.
- Series convergence: Σ a_n converges if the partial sums s_n = Σ from k=1 to n of a_k have a finite limit.
- Partial sum formula: s_n = a_1 + a_2 + ... + a_n.
- Necessary condition for series convergence: if Σ a_n converges, then lim n→∞ a_n = 0.
- The converse is false: lim n→∞ a_n = 0 does not always mean Σ a_n converges.
- Geometric series rule: Σ ar^n converges when |r| < 1 and diverges when |r| ≥ 1.
Vocabulary
- Sequence
- A sequence is an ordered list of terms a_1, a_2, a_3, and so on.
- Series
- A series is the sum of the terms of a sequence, written as Σ a_n.
- Convergence
- Convergence means that a sequence or the partial sums of a series approach a finite limit.
- Partial Sum
- A partial sum is the sum of the first n terms of a series, written s_n = a_1 + a_2 + ... + a_n.
- Divergence
- Divergence means that no finite limit is approached, either because values grow, oscillate, or fail to settle.
Common Mistakes to Avoid
- Thinking a_n → 0 automatically makes Σ a_n converge. This is wrong because terms getting small is necessary but not sufficient, as the harmonic series Σ 1/n diverges.
- Confusing the terms a_n with the partial sums s_n. A sequence tracks individual entries, while a series tracks accumulated totals.
- Testing a series by only finding lim n→∞ a_n and stopping. If the limit is not zero the series diverges, but if the limit is zero more tests are still needed.
- Assuming bounded terms imply a convergent series. A series depends on the behavior of cumulative sums, so even bounded terms like a_n = 1 make partial sums grow without bound.
Practice Questions
- 1 For the sequence a_n = 5 + 2/n, find lim n→∞ a_n and state whether the sequence converges.
- 2 For the series Σ from n=1 to ∞ of (1/3)^n, determine whether it converges and find its sum.
- 3 A student says that since 1/n approaches 0, the series Σ from n=1 to ∞ of 1/n must converge. Explain the error in this reasoning.