Determine the sum of the geometric series: 4 + 2 + 1 + 1/2 + \dots.

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{"type":"doc","content":[{"content":[{"marks":[{"type":"bold"}],"text":"Step 1 (Identify Series Type):","type":"text"},{"type":"text","text":" The given series is "},{"attrs":{"latex":"4 + 2 + 1 + 1/2 + \dots"},"type":"math_inline"},{"type":"text","text":". This is an infinite geometric series where:"}],"type":"paragraph"},{"content":[{"content":[{"content":[{"type":"text","text":"The first term is "},{"attrs":{"latex":"a = 4"},"type":"math_inline"}],"type":"paragraph"}],"type":"list_item"},{"content":[{"type":"paragraph","content":[{"type":"text","text":"The common ratio is "},{"attrs":{"latex":"r = \frac{2}{4} = \frac{1}{2}"},"type":"math_inline"}]}],"type":"list_item"}],"type":"bullet_list"},{"content":[{"marks":[{"type":"bold"}],"text":"Step 2 (Verify Convergence):","type":"text"},{"text":" The common ratio satisfies ","type":"text"},{"attrs":{"latex":"|r| = 0.5 < 1"},"type":"math_inline"},{"text":", which means the infinite geometric series converges to a finite sum.","type":"text"}],"type":"paragraph"},{"content":[{"marks":[{"type":"bold"}],"text":"Step 3 (Apply Sum Formula):","type":"text"},{"text":" The sum ","type":"text"},{"type":"math_inline","attrs":{"latex":"S"}},{"text":" of an infinite geometric series is given by:","type":"text"}],"type":"paragraph"},{"attrs":{"latex":"S = \frac{a}{1 - r}"},"type":"math_block"},{"attrs":{"latex":"S = \frac{4}{1 - 1/2} = \frac{4}{1/2} = 4 \times 2 = 8"},"type":"math_block"},{"content":[{"marks":[{"type":"bold"}],"text":"Step 4 (Conclusion):","type":"text"},{"type":"text","text":" The sum of the geometric series is exactly "},{"text":"8","type":"text","marks":[{"type":"code"}]},{"text":".","type":"text"}],"type":"paragraph"}]}

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