My textbook defines the renewal function $R(t) = E[N_t] = \sum_{n=0}^\infty F^n(t)$, where $F^n(t)$ appears to be the n-fold convolution of $F$ with itself. $F$ is the distribution of the interrenewal times in a renewal process. I'm not sure how to actually calculate this though.
I'm kind of struggling with the idea of convolution, which might be why I can't figure this out.
One specific example gives $F(t) = 1 - e^{-3t} - 3te^{-3t}$. I'm unsure how to find $R(t)$ given this though. $R = 1 + F + F^2 + F^3 + \cdots$, but the definition of convolution seems too complex to me for there to be a good direct way to calculate this sum. Is there any easier way to solve for $R$?
Thanks.
EDIT:
... I still haven't figured this out. If I use the renewal equation, $R = F + R*F$, in this example, I have:
$R = 1 - e^{-3t} -3te^{-3t} + (R * (1 - e^{-3t} -3te^{-3t}))$
$R = 1 - e^{-3t} -3te^{-3t} + \int_0^t{F(t-x)dR(x)}$ by the definition of convolution.
$R = 1 - e^{-3t} -3te^{-3t} + \int_0^t{(1-e^{-3(t-x)}-3(t-x)e^{-3(t-x)})dR(x) }$
But I can't calculate this integral since I don't know what $R(x)$ is. At this point, I was thinking of differentiating with respect $R(x)$ but I'm not sure if this would actually work.