Counterexample bank

Part C is won by knowing which tempting claims are false. 148 counterexamples; 85 free. The rest come with the Notes pack.

#1 · Analysis & Linear Algebra › The Real Line

“Σaₙ convergent ⇒ Σaₙ² convergent” — false

Counterexample locked — unlock with Notes + PYQ

series

#2 · Analysis & Linear Algebra › The Real Line

“aₙ → 0 ⇒ Σaₙ converges” — false

Counterexample: Harmonic series Σ1/n

series

#3 · Analysis & Linear Algebra › The Real Line

“Σaₙ converges ⇒ Σaₙ² converges” — false

Counterexample: aₙ = (−1)ⁿ/√n

Alternating series converges; squares give the harmonic series.

series

#4 · Analysis & Linear Algebra › The Real Line

“Ratio test inconclusive ⇒ root test inconclusive” — false

Counterexample locked — unlock with Notes + PYQ

series

#5 · Analysis & Linear Algebra › Sequences and Series of Functions

“Σaₙxⁿ → L as x → 1⁻ ⇒ Σaₙ = L” — false

Counterexample: Σ(−1)ⁿxⁿ = 1/(1 + x) → 1/2

$\sum (- 1)^{n}$ diverges. Abel's theorem has no converse without a Tauberian condition.

seriespower series