Series Explorer: Three Views of Infinity
Watch how partial sums behave as you add more terms. Does the sum settle down or grow forever?
Choose a Series
Partial Sum:
0.000
Behavior:
Settles
Formula:
Σ rⁿ
➡️ Number Line — Watch the Sum Accumulate
📐 The Necessary Condition
If Σaₙ converges, then aₙ → 0
For a series to have a finite sum, its terms must eventually become negligible.
✓ Terms shrink to 0 → ✓ Sum settles at 2
🚨 But The Converse is FALSE!
aₙ → 0 does NOT guarantee convergence
The harmonic series is the classic counterexample: each term 1/n → 0, yet Σ(1/n) → ∞
✓ Terms shrink to 0 → ✗ Sum STILL grows forever!
🧪 The Divergence Test
If an does not → 0, then Σan diverges
This is the contrapositive of the necessary condition — and it's always conclusive!
Terms: aₙ = n/(n+1) → 1 ≠ 0
✗ Terms don't go to 0 → Series MUST diverge (no further analysis needed)
⚠️ Caution: If aₙ → 0, the test is inconclusive. The series might converge (like geometric) or diverge (like harmonic). We need other tools!
⚖️ The Comparison Principle
When the divergence test is inconclusive (aₙ → 0), comparison can help. Click each case:
📉
Smaller than Convergent
Must converge!
📈
Bigger than Divergent
Must diverge!
🤷
Bigger than Convergent
Could go either way!
😕
Smaller than Divergent
Could go either way!
MATHSWELL