Step 1 :Express the sum using summation notation: \(\sum_{k=0}^{8}(-1)^{k}\left(\frac{1}{3^{k}}\right)\)
Step 2 :Complete the sum of the sequence using the formula for the sum of a geometric series.
Step 3 :Calculate the sum: \(\sum_{k=0}^{8}(-1)^{k}\left(\frac{1}{3^{k}}\right)\)
Step 4 :Final Answer: The sum of the series is \(\boxed{0.7500381039475692}\)