Practice section 7 of 17

In the source legend, a chess inventor asks for one grain of rice on the first square, two on the second, four on the third, eight on the fourth, and twice as many on each following square. How many grains are required in total for all $64$ squares?

Find the sum of the first $15$ terms of a GP with first term $a=0.6$ and common ratio $r=1.5$.

Let $S_n$ be the sum of the first $n$ terms of $1+2+4+8+16+\cdots$. Find the least positive integer $n$ such that $S_n>2019$.

Express each series in sigma notation and find its sum: (a) $3+6+12+24+48+96$; (b) $\frac12+\frac14+\frac18+\frac1{16}+\cdots+\frac1{512}$.

Given that $\displaystyle\sum_{n=1}^{N}\frac1{n(n+1)(n+2)}=\frac14-\frac1{2(N+1)}+\frac1{2(N+2)}$, explain whether the series is convergent or divergent, and state the sum to infinity if it is convergent.

Classify the series represented by these partial sums as convergent or divergent, giving the finite limit when one exists. (a) $\sum_{j=1}^{n}2^j=2(2^n-1)$. (b) Given $\sum_{j=1}^{n}\frac1{j(j+2)}=\frac34-\frac1{2(n+1)}-\frac1{2(n+2)}$. (c) An AP with $S_n=\frac n2[2a+(n-1)d]$, for $d>0$ and $d<0$. (d) A GP with $S_n=\frac{a(r^n-1)}{r-1}$, $r>1$, $a\ne0$. (e) A GP with $S_n=\frac{a(1-r^n)}{1-r}$, $|r|<1$.