Series Convergence
Grade 12 · Math · Study Guide
Determine whether infinite series converge or diverge using key tests.
An infinite series is the sum of infinitely many terms: Σaₙ from n=1 to ∞. A series converges if the partial sums Sₙ approach a finite limit L; otherwise it diverges. The nth-term test states: if lim aₙ ≠ 0, the series diverges.
A geometric series Σarⁿ converges to a/(1-r) when |r| < 1 and diverges when |r| ≥ 1. A p-series Σ1/nᵖ converges when p > 1 and diverges when p ≤ 1.
The Ratio Test is powerful for series involving factorials or exponentials: compute L = lim|aₙ₊₁/aₙ|. If L < 1 the series converges absolutely; if L > 1 it diverges; if L = 1 the test is inconclusive.
| Test | When to Use | Converges If |
|---|---|---|
| nth-Term | Always try first | lim aₙ = 0 (necessary, not sufficient) |
| Geometric | Series has rⁿ form | |r| < 1 |
| p-Series | Series is Σ1/nᵖ | p > 1 |
| Ratio | Factorials or exponentials | lim|aₙ₊₁/aₙ| < 1 |
| Integral | aₙ = f(n) continuous, positive, decreasing | ∫₁^∞ f(x) dx converges |
| Condition | Result |
|---|---|
| |r| < 1 | Converges; sum = a/(1-r) |
| |r| ≥ 1 | Diverges |
Key terms
- Convergence
- A series converges if its partial sums approach a finite limit.
- Divergence
- A series diverges if its partial sums grow without bound or oscillate.
- Geometric series
- Σarⁿ — converges to a/(1-r) when |r| < 1.
- Ratio Test
- Compare |aₙ₊₁/aₙ| as n→∞; L<1 converges, L>1 diverges.
Practice questions
Does the geometric series Σ (1/3)ⁿ (n=0 to ∞) converge?
- No, it diverges
- Yes, sum = 3/2
- Yes, sum = 1/2
- Yes, sum = 3
Show answer
a=1, r=1/3. |r|<1 so converges. Sum = 1/(1-1/3) = 1/(2/3) = 3/2.
Does the p-series Σ 1/n² converge?
- Yes
- No
- Only partially
- Cannot determine
Show answer
p = 2 > 1, so the p-series converges.
Apply the nth-term test to Σ n/(n+1). Does it converge or diverge?
Show answer
lim n/(n+1) = 1 ≠ 0. Since the nth term doesn't approach 0, the series diverges.
If lim aₙ = 0, the series Σaₙ must converge.
Show answer
The harmonic series Σ1/n has lim 1/n = 0 but diverges. lim aₙ = 0 is necessary but not sufficient.
Use the Ratio Test on Σ n!/2ⁿ. What can you conclude?
- Converges
- Diverges
- Inconclusive
- Sum = 2
Show answer
L = lim|(n+1)!/2ⁿ⁺¹ · 2ⁿ/n!| = lim(n+1)/2 = ∞ > 1. Diverges.
Fun facts
- The harmonic series 1 + 1/2 + 1/3 + 1/4 + ... diverges, even though its terms shrink to 0 — it just grows infinitely slowly.
- Zeno's paradox (Achilles and the tortoise) is resolved by a convergent geometric series: the infinite sum of distances is finite.
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