Discussion questions · Section 2 of 3

The series $1+e^{-x}+e^{-2x}+\cdots$ has a sum to infinity. Show that $x>0$. When $x=10$, write $S_n$ and $S$, the finite and infinite sums. Show that $S-S_n<S/10^{100}$ requires $n>10\ln10$.

The seventh, third and first AP terms (nonzero common difference) are respectively the first three GP terms. Show that the geometric series converges. The seventh AP term is $3$. Find the smallest $n$ for which the first $n$ AP terms exceed the first $n$ GP terms by at least $100$.

Find (i) $\sum_{k=0}^n(k+n+1)$ in terms of $n$; (ii) $\sum_{r=1}^N(4r+2^{-r})$ in terms of $N$; (iii) $\sum_{k=1}^{30}(a^{2k}+ka)$ in terms of $a$.

Given $\sum_{r=1}^nr(r+2)=n(n+1)(2n+7)/6$, find $\sum_{r=n}^{2n}r(r+2)$.

Given $\sum_{r=1}^n\dfrac{7r+10}{r(r+1)(r+2)}=6-\left(\dfrac5{n+1}+\dfrac2{n+2}\right)$, deduce the exact value of $\sum_{r=22}^\infty\dfrac{21r+30}{r(r+1)(r+2)}$.

Given $\sum_{r=2}^n\dfrac{3r^2+1}{(r^2-1)^3}=\dfrac12\left[\dfrac98-\dfrac1{n^3}-\dfrac1{(n+1)^3}\right]$, find $\sum_{r=3}^n\dfrac{3(r-1)^2+1}{[(r-1)^2-1]^3}$ in terms of $n$. Hence find its infinite sum.