Find a power series representation centered at 0 for the following function using known power series. Give the interval of convergence for the resulting series.
\[
f(x)=\frac{19}{19+x}
\]
Which of the following is the power series representation for $f(x)$ ?
A. $\sum_{k=0}^{\infty}\left(-x^{19}\right)^{k}$
B. $\sum_{k=0}^{\infty} 19 x^{k}$
C. $\sum_{k=0}^{\infty}(-19 x)^{k}$
D. $\sum_{k=0}^{\infty}\left(-\frac{x}{19}\right)^{k}$

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