Questions: Series Definition and Partial Sums

Question 1

The series Σ (1/n) — the harmonic series — has terms that approach zero as n → ∞. Does it converge?

• A. Yes — since the terms go to zero, the partial sums must stabilize
• B. No — the terms go to zero too slowly for the partial sums to converge
• C. Yes — any series whose terms shrink to zero converges by definition
• D. It depends on where you start the sum

Answer: (B)

The harmonic series diverges even though its terms approach zero. Terms going to zero is necessary but not sufficient for convergence. The partial sums of Σ(1/n) grow without bound — they just grow very slowly (logarithmically). This is the key asymmetry: if terms don't go to zero, the series definitely diverges, but the converse is false. The harmonic series is the canonical counterexample every calculus student must internalize.

Question 2

For the series Σ aₙ, the notation S_N refers to:

• A. The Nth term of the series, aₙ evaluated at n = N
• B. The sum of the first N terms: a₁ + a₂ + ⋯ + aₙ
• C. The limit of the series as it approaches its sum S
• D. The number of terms needed for the partial sum to exceed N

Answer: (B)

S_N is the Nth partial sum — a finite sum of the first N terms. This is a concrete, computable number. The series itself is defined as lim(N→∞) S_N. The confusion between aₙ (individual terms) and S_N (accumulated total) is one of the most common errors in this topic: aₙ is what you're adding; S_N is what you've added so far.

Question 3

Saying that an infinite series 'converges to S' is really a statement about a sequence of partial sums converging to S.

• T. True
• F. False

Answer: True

This is exactly the definition. The series Σ aₙ converges to S means the sequence {S_N} — where S_N = a₁ + ⋯ + aₙ — converges to S as N → ∞. This is not just a reformulation; it is the precise definition that makes infinite sums mathematically rigorous by reducing them to limits of sequences, which you already know how to handle.

Question 4

If the terms of a series approach zero, the series should converge.

• T. True
• F. False

Answer: False

This is the most persistent misconception in infinite series. The harmonic series Σ 1/n is the canonical counterexample: each term 1/n → 0, but the partial sums grow without bound. Terms approaching zero is a necessary condition for convergence — if terms don't go to zero, the series definitely diverges — but it is not sufficient. The terms must shrink fast enough that their cumulative sum stays bounded.

Question 5

Why can't we define an infinite series simply as 'the result of adding infinitely many numbers together,' the way we add finitely many numbers?

Model answer: Addition is a binary operation — it takes two inputs. Finite sums extend this by repeating the operation, but 'infinitely many additions' cannot be completed in any finite number of steps; there is no last step that yields a result. The definition via partial sums resolves this by converting the infinite process into a limit of finite computations: S_N = a₁ + ⋯ + aₙ is always well-defined, and we define the series as lim S_N.

Without this definition, 'infinite sum' is meaningless. The brilliance of the partial sums definition is that it reduces a new concept (infinite series) to a concept already understood (limits of sequences), giving access to all the tools of sequence analysis. Every convergence test you will learn is ultimately a test for whether the sequence {S_N} converges.