Infinite sequences & series
The idea
An infinite series asks whether adding infinitely many numbers can settle on a finite total. The answer lives in partial sums: the series converges exactly when the sequence of partial sums has a limit. You have summed finite geometric and arithmetic progressions before; the university-level question is qualitative first — converge or diverge? — because most convergent series have no closed-form sum, and the convergence verdict alone is what later work (power series, Fourier series, probability) depends on.
Think of the convergence tests as a triage protocol. First, the nth-term test: if the terms do not tend to 0, the series diverges immediately. Then match the series to a template: geometric (converges when the common ratio satisfies |r| < 1), p-series (converges when p > 1), comparison against a known series, the ratio test for factorials and exponentials, and the alternating series test for sign-flipping terms.
The single most damaging misconception: terms shrinking to zero is necessary but nowhere near sufficient. The harmonic series 1 + 1/2 + 1/3 + … has terms tending to 0 yet diverges — its partial sums creep past every bound. Shrinking terms make convergence possible; only shrinking fast enough makes it actual.
Worked example
Determine whether the series with terms a_n = n²/3^n (summed from n = 1 to infinity) converges or diverges.
- Triage: the terms mix a polynomial n² against an exponential 3^n, and exponential-versus-polynomial contests are the ratio test's specialty.
- Form the ratio of consecutive terms: a_(n+1)/a_n = ((n + 1)²/3^(n+1)) × (3^n/n²) = (1/3) × ((n + 1)/n)².
- Take the limit as n → ∞: ((n + 1)/n)² → 1, so the ratio tends to L = 1/3.
- Apply the ratio test verdict: L = 1/3 < 1, so the series converges absolutely — the exponential denominator eventually crushes the polynomial growth in the numerator.
- Sanity-check with early terms: 1/3, 4/9, 9/27 = 1/3, 16/81 ≈ 0.198, 25/243 ≈ 0.103 — after a brief rise the terms decay roughly geometrically, consistent with the test.
Answer. The series of n²/3^n converges absolutely by the ratio test, since the limiting ratio 1/3 is less than 1.
Check your understanding
- Why does the harmonic series diverge even though its terms tend to zero?
- How do you decide which convergence test to try first when handed an unfamiliar series?
- What does the ratio test's limit L actually measure about the long-run behavior of the terms?
- Why does the ratio test return no verdict when L = 1, and what kind of series typically lands there?
Build the foundations first
Infinite sequences & series builds on these concepts. If any feel shaky, start there.