I am working on this renewal theory problem.
Suppose that we are a character in a video game collecting coins. Let $X_n, n \geq 1$ be the time between the $(n - 1)$th and $n$th coin collected, where each $X_n$ has probability density function
$$ f(x) = \begin{cases} \frac{2\sqrt{2}}{\pi(x^4 + 1)}, & x > 0 \\ 0, & x \leq 0. \end{cases} $$
Let $S_n = \sum_{i = 1}^n X_i$. Thus, $S_n$ gives the time we collect the $n$th coin. Let $N(t)$ be the number of coins collected by time $t$. Approximate the probability that we collect at least $230$ coins in $3$ hours.
I am working with Probability Models by Sheldon Ross as my text. So far, I think that we have
\begin{align} P(N(3) \geq 230) & \iff P(S_{230} \leq 3) \\ & \iff P(X_1 + \cdots + X_{230} \leq 3). \end{align}
We want to calculate $P(X_1 + \cdots + X_{230} \leq 3)$. I think this is supposed to be the n-fold convolution of the probability density, but I do not know how to do that. Any help would be appreciated.