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Evaluate the limit: lim(n->inf) of the sum from r=1 to n of arctan( 2*3^(r-1) / (1 + 3^(2r-1))).

1. 3*pi/4
2. cot⁻¹(3)
3. tan⁻¹(3)
4. pi/4

Correct answer: pi/4

Solution

The general term equals arctan(3^r) - arctan(3^(r-1)), so the sum telescopes to arctan(3ⁿ) - arctan(1). As n->inf, arctan(3ⁿ) -> pi/2 and arctan(1) = pi/4, giving the limit pi/2 - pi/4 = pi/4.

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