Sequences, Geometric Series and Telescoping Sums
- Distinguish a sequence from a series and the convergence of each
- Decide whether a geometric series converges and find its sum when it does
- Evaluate a telescoping series by examining its partial sums
A series converges when its partial sums converge
A sequence is a list of terms aₙ; a series is the sum of that list. The series Σaₙ converges when the sequence of partial sums Sₙ = a₁ + a₂ + … + aₙ approaches a finite limit, and the value of the series is that limit. This distinction drives the whole unit: the harmonic series has terms 1/n going to zero while its partial sums grow without bound, so the sequence converges and the series does not.
Geometric series are the one family with a formula
Σ (n = 0 to ∞) arⁿ converges if and only if |r| < 1, and then its sum is a/(1 − r), where a is the first term actually present. Two details decide most of the errors: r is the ratio between consecutive terms, and a is whatever the first term is — if the sum starts at n = 2, then a is that term, not the n = 0 term. Geometric series matter beyond their own questions because the ratio test, power series and the Maclaurin series for 1/(1 − x) are all built on them.
The nth-term test can only prove divergence
If lim aₙ ≠ 0, the series diverges. The converse is false and is the most misused statement in the unit: aₙ → 0 tells you nothing, because both the convergent Σ1/n² and the divergent Σ1/n have terms going to zero. So the nth-term test is a fast first screen — if the terms do not vanish, stop, the series diverges — and never a proof of convergence.
Evaluate Σ (n = 1 to ∞) of 1/(n(n + 1)) by examining its partial sums.
- 1.Decompose by partial fractions: 1/(n(n + 1)) = 1/n − 1/(n + 1).
- 2.Write out the partial sum: Sₙ = (1 − 1/2) + (1/2 − 1/3) + (1/3 − 1/4) + … + (1/n − 1/(n + 1)).
- 3.Every interior term cancels with its neighbor — the sum telescopes, leaving only the first and last pieces.
- 4.So Sₙ = 1 − 1/(n + 1).
- 5.Take the limit: lim(n→∞) [1 − 1/(n + 1)] = 1. The series converges to 1.
Telescoping and geometric are the only two families where the exam expects an exact sum. For everything else the question is whether the series converges, not what it converges to.
What is the sum of Σ (n = 2 to ∞) of 3·(1/4)ⁿ?
A series has terms with lim(n→∞) aₙ = 0. What follows?
Writing "aₙ → 0, therefore the series converges" is scored as a wrong statement even when the series does happen to converge. The rubric penalizes the reasoning, not just the conclusion.
Answer the 2 checkpoints as you read.
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