## $x$ is within 3 units of $c$ February 14

I want to use absolute value notation to represent the phrase "$x$ is within 3 units of $c$". Which of the following is the correct answer? (i) or (ii)? (i) $|x-c|<3$, (ii) $|x-c|\leq{3}$

## Finite Geometric Series Proof February 14

Prove the Riemann Sum to the $n$th term with $i=1$ of $ar^{i-1}$ is equivalent to $a\frac{r^n -1}{r-1}$. This is true for a geometric series when $r$ doesn't equal $1$.

## expected and renewal process February 14

Let $N_t$ a renewal process with $T_i$ the jumps. I know $U(t)=E(N_t)$ on [0,t]. Let $B_t=T_{N_t}-t$ and $G_t(u)=P(B_t \leq u)$How to show on ]t,t+a]: $E(N_{t+a}-N_t)=\int_0^a U(a-u)G_t(du)=(G_t \star U)(a)=(U \star G_t)(a)$Thank you