Question

find the infinite sum, if it exists: \\(-2 + (0.5) + (-0.125) + \cdots\\)

Explanation:

Step1: Identify the first term

$$a_1 = -2$$

Step2: Calculate the common ratio

$$r = \frac{0.5}{-2} = -0.25$$

Step3: Check for convergence

$$|r| = 0.25 < 1$$

Step4: Apply the sum formula

$$S = \frac{a_1}{1 - r} = \frac{-2}{1 - (-0.25)}$$

Step5: Simplify the expression

$$S = \frac{-2}{1.25} = -1.6$$

Answer:

\(-\frac{8}{5}\) (or \(-1.6\))