The infinite series is given.

MATHEMATICS · Multi-part · A Level · Mathematics · Sequences and series · Sequences and series

Problem

The infinite series 23−23+233−293+⋯2\sqrt{3} - \dfrac{2}{\sqrt{3}} + \dfrac{2}{3\sqrt{3}} - \dfrac{2}{9\sqrt{3}} + \cdots is given. (i) Determine the common ratio of the corresponding geometric progression and state whether the series is convergent. (ii) If the series is convergent, calculate its sum to infinity. (iii) Write the numerical value of the sum obtained in part (ii) correct to three decimal places.

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