Practice section 8 of 17

What happens to the infinite series when (a) $d=0$ in an AP, and (b) $r=1$ in a GP?

Use limit notation to rewrite: (a) as $n\to\infty$, $u_n=(\frac12)^{n-1}\to0$; (b) $S_n\to2$ as $n\to\infty$, where $S_n=1+\frac12+\frac14+\cdots+(\frac12)^{n-1}$.

For the geometric partial sum $S_n=\frac{a(1-r^n)}{1-r}$, consider what happens to $r^n$ as $n\to\infty$ and whether there is a finite sum to infinity. Assume $a\ne0$; treat $r=1$ separately from this formula.

Does an infinite arithmetic series have a sum to infinity? Why or why not?

For each geometric progression, determine the sum to infinity, if it exists: (i) $3,9,27,81,\ldots$; (ii) $0.9,0.9^2,0.9^3,0.9^4,\ldots$; (iii) $2,-\frac43,\frac89,-\frac{16}{27},\ldots$.

The second term of a geometric progression is $5$ while its sum to infinity is $20$. Find the sum of the first $8$ terms.