Series & convergence basics
An infinite series converges when its partial sums approach a limit. The first check is always the divergence test: if the terms do not tend to zero, the series diverges. But terms tending to zero is not enough — the harmonic series Σ1/n diverges despite 1/n → 0, which is why the comparison, ratio and integral tests exist.
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Method: how to approach it
The order below is what examiners expect to see, and each step carries its own marks.
- Apply the divergence test firstIf aₙ ↛ 0 the series diverges immediately. If aₙ → 0 the test is inconclusive and you continue.
- Recognise the standard familiesGeometric Σarⁿ converges iff |r| < 1; p-series Σ1/n^p converges iff p > 1.
- Use the ratio test for factorials and powersIf lim |aₙ₊₁/aₙ| = L, the series converges absolutely for L < 1 and diverges for L > 1. L = 1 is inconclusive.
- Compare with a known seriesDirect or limit comparison against a p-series or geometric series settles most remaining cases.
Worked example
Does Σ_{n=1}^∞ n/2ⁿ converge?
- Terms tend to 0, so the divergence test is inconclusive.
- Ratio test: aₙ₊₁/aₙ = ((n+1)/2^(n+1)) · (2ⁿ/n) = (n+1)/(2n).
- As n → ∞ this tends to 1/2.
- 1/2 < 1, so the series converges absolutely.
Answer. It converges — and in fact sums to 2, since Σ n xⁿ = x/(1−x)² at x = 1/2.
Where marks get dropped
These are the specific errors that cost credit on series & convergence basics questions — QED's rubric penalises each of them separately.
- Concluding convergence because aₙ → 0. The harmonic series is the standard counterexample: terms vanish but the sum is infinite.
- Using the ratio test when L = 1 and calling it divergent. That case gives no information — try comparison instead.
- Confusing a sequence with a series. The sequence 1/n converges to 0 while the series Σ1/n diverges.
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Series & convergence basics — frequently asked questions
Why does the harmonic series diverge?
Group terms as 1/3 + 1/4 > 1/2, 1/5 + … + 1/8 > 1/2, and so on. Infinitely many blocks each exceeding 1/2 give an unbounded sum.
What is absolute convergence?
Σ|aₙ| converges. It implies convergence and permits rearranging terms freely — conditionally convergent series can be rearranged to any sum at all.
When should I use the integral test?
When aₙ = f(n) for a positive decreasing continuous f that you can integrate. It is how the p-series result is proved.
The rest of Calculus
Limits, derivatives, integrals and their applications. Each subtopic below has its own method, worked example and mark-losing traps.
- 1Limits & continuity
- 2Product, quotient & chain rules
- 3Tangent lines & linear approximation
- 4Extrema & optimisation
- 5Indefinite & definite integrals
- 6Substitution & integration by parts
- 7Series & convergence basics
- 8Implicit differentiation & related rates
- 9Mean value & intermediate value theorems
- 10Improper integrals & convergence
- 11Taylor & Maclaurin polynomials
- 12L’Hôpital’s rule & indeterminate forms
- 13Curve sketching: asymptotes, concavity & inflection
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