Final answer:
The sum of the series is -2.002, which can be represented as a repeating decimal as -2.002002002... and as a fraction in lowest terms as -1001/500.
Step-by-step explanation:
To find the sum of the series ∑I=1[∞]2(1000 / 1)ᶦ, we can rewrite it as ∑I=1[∞]2(10^3)ᶦ. Using the formula for the sum of a geometric series, we have:
S = a / (1 - r), where a is the first term and r is the common ratio. In this series, a = 2000 and r = 1000. Substitute the values into the formula:
S = 2000 / (1 - 1000), S = 2000 / (-999). The sum of the series is -2.002, which can be represented as a repeating decimal as -2.002002002...
As a fraction in lowest terms, we can simplify -2.002 to -2002/1000 = -1001/500.