∑aₙ Diverges — Does ∑aₙ/(1+naₙ) Converge? The Taming Trick That ALWAYS Fails

📢 Question of the Day | Series Transformation — VERY HARD
🎯 Topic: bₙ = aₙ/(1+naₙ) always diverges when ∑aₙ diverges
📅 CSIR NET Part C Level | GATE Mathematics | IIT JAM

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∑aₙ diverges. Now define bₙ = aₙ/(1 + naₙ).
The denominator looks like it "tames" aₙ and forces
convergence of ∑bₙ. Is this intuition correct?

Examples verified: aₙ = 1/n, 1/n^2

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