Section 1
Infinite series
Show that$$\frac{1}{r !}-\frac{1}{(r+1) !}=\frac{r}{(r+1) !} \text { for } r \in \mathbb{N}$$Hence determine the $n$th partial sum of $\sum_{r=1}^{\infty} r /(r+1) !$ and show that$$\sum_{r=1}^{\infty} \frac{r}{(r+1) !}=1$$
Use the vanishing condition $(4.1 .2)$ to show that cach of the following scrics is divergent:(a) $\sum_{r=1}^{\infty} \frac{r}{r+1}$(b) $\sum_{r=1}^{\infty}[r-\sqrt{r(r-1)}]$
Prove the sum and scalar product rules for series (see 4.1.3 and 4.1.4).
Suppose that $\sum_{r=1}^{\infty} a_{r}$ is convergent and $\sum_{r=1}^{\infty} b_{r}$ is divergent. Prove that $\sum_{r=1}^{\infty}\left(a_{r}+b_{r}\right)$ is divergent.