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Consider the following statements: Statement I: \(\tan^{-1}(1/3) + \tan^{-1}(1/2) + \tan^{-1}(1/4) + \cdots = \pi/4\) Statement II: Whenever \(xy < 1\), we have \(\tan^{-1}x + \tan^{-1}y = \tan^{-1}\!\left(\frac{x+y}{1-xy}\right)\). Choose the correct option.

1. Statement I is correct, Statement II is correct, and Statement II correctly explains Statement I.
2. Statement I is correct, Statement II is correct, but Statement II does not correctly explain Statement I.
3. Statement I is incorrect, while Statement II is correct.
4. Statement I is correct, while Statement II is incorrect.

Correct answer: Statement I is correct, Statement II is correct, but Statement II does not correctly explain Statement I.

Solution

Statement I is correct as the series converges to C0/4, while Statement II is a valid identity for the sum of inverse tangents when the product of the arguments is less than one, but it does not directly explain the convergence of the series in Statement I.

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