Step 1 :Rewrite the function \(f(x)=\frac{19}{19+x}\) as \(f(x)=19(1+x/19)^{-1}\).
Step 2 :Recognize this as a geometric series with first term \(a=19\) and common ratio \(r=-x/19\).
Step 3 :The power series representation of a geometric series is \(\sum_{k=0}^{\infty} ar^{k}\).
Step 4 :Substitute \(a=19\) and \(r=-x/19\) into the formula to get the power series representation of \(f(x)\): \(\sum_{k=0}^{\infty} 19(-x/19)^{k}\).
Step 5 :The interval of convergence for a geometric series is \(|r|<1\).
Step 6 :Therefore, the interval of convergence for this series is \(|-x/19|<1\), which simplifies to \(-19 Step 7 :Final Answer: The power series representation for \(f(x)\) is \(\boxed{\sum_{k=0}^{\infty}\left(-\frac{x}{19}\right)^{k}}\) and the interval of convergence is \(\boxed{-19