Final answer:
The question deals with mathematical series expansions and properties of exponents, touching upon a geometric series, probability distribution functions, and the central limit theorem. The error provided in the CDF evaluation (P(X< 1) = 1) is corrected to approximately 0.3935.
Step-by-step explanation:
The original question provided relates to the properties of series expansions in mathematics, specifically regarding the geometric series and its application. The given formula 1/(1 - x) = Σ[[infinity], n=0] xⁿ for |x| < 1 represents a geometric series where each term is x raised to the power of n. Now, let's clarify the related information provided:
The formula X~ Exp(0.5) signifies that the random variable X follows an exponential distribution with a rate-parameter of 0.5. The related cumulative distribution function (CDF) P(X < x) = 1 – e−(0.5)x and the probability P(X< 1) given by 1 - e−(0.5) is approximately 0.3935, not equal to 1 as stated.
Additionally, we have information regarding exponentiation rules and manipulating expressions with exponents, such as the idea that dividing by a term with an exponent is equivalent to negating the exponent. For example, x−n is equal to 1/xn.
Finally, we have an example regarding a discrete probability distribution function (PDF) where the sum of the probabilities equals one, as required for any valid PDF. The central limit theorem is also referenced with the formula EX ~ N[(80)(90), (√80)(15)], indicating that a sum of random variables can be approximated by a normal distribution given a significant number of samples (n = 80 in this case).