The function $f(x)=\frac{5}{1+30 x^{2}}$ is represented as a power series
\[
f(x)=\sum_{n=0}^{\infty} c_{n} x^{n} .
\]

Find the first few coefficients in the power series.
\[
\begin{array}{l}
c_{0}=\square \\
c_{1}=\square \\
c_{2}=\square \\
c_{3}=\square
\end{array}
\]

Find the radius of convergence $R$ of the series
\[
R=\square \text {. }
\]

Step 1 ：The function $f(x)=\frac{5}{1+30 x^{2}}$ is represented as a power series $f(x)=\sum_{n=0}^{\infty} c_{n} x^{n}$. The coefficients can be found by differentiating the function and evaluating at $x=0$.

Step 2 ：To find $c_0$, we simply evaluate the function at $x=0$. $c_0 = 5$.

Step 3 ：To find $c_1$, we differentiate the function once and evaluate at $x=0$. $c_1 = 0$.

Step 4 ：To find $c_2$, we differentiate the function twice and evaluate at $x=0$. $c_2 = -300$.

Step 5 ：To find $c_3$, we differentiate the function three times and evaluate at $x=0$. $c_3 = 0$.

Step 6 ：The coefficients $c_0, c_1, c_2, c_3$ are 5, 0, -300, and 0 respectively.

Step 7 ：The radius of convergence of a power series is the value of $x$ for which the series converges. It can be found using the ratio test, which states that if the limit as $n$ approaches infinity of the absolute value of the ratio of the $(n+1)$th term to the $n$th term is less than 1, then the series converges.

Step 8 ：Since the coefficients $c_n$ are alternating between 0 and a non-zero value, the limit on the left side is 0 for all $n$. Therefore, the inequality holds for all $x$, and the radius of convergence is infinity.

Step 9 ：The final answer is: The coefficients $c_0, c_1, c_2, c_3$ are \(\boxed{5}\), \(\boxed{0}\), \(\boxed{-300}\), and \(\boxed{0}\) respectively. The radius of convergence $R$ is \(\boxed{\infty}\).