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Evaluate the limit: lim (n -> infinity) 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 key identity is arctan(x) - arctan(y) = arctan((x-y)/(1+xy)) when xy > -1. Setting x = 3^r and y = 3^(r-1), the numerator is 3^r - 3^(r-1) = 2 * 3^(r-1) and the denominator is 1 + 3^(2r-1), which matches the given expression exactly. The sum therefore telescopes to arctan(3ⁿ) - arctan(3⁰). Taking the limit gives pi/2 - pi/4 = pi/4.

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