This question concerns the infinite geometric series formula. It turns out there is a nice formula for the sum of an infinite geometric series.
Consider the infinite geometric series $1+r+r^2+r^3+\cdots$ For this series to converge, what must be the true about $r$? Explain.
I have totally no clue what it means. Can anyone help me with this question? Thank you very much.
Ok, so
$$\sum_{k=0}^n r^k=\frac{1-r^n}{1-r}\;,\;\;r\neq 1$$
and this is true always. Now
$$\lim_{n\to\infty}\frac{1-r^n}{1-r}=\begin{cases}\frac1{1-r}&,\;\;|r|<1\\{}\\\pm\infty&,\;\;|r|>1\end{cases}$$
and from the above is clear the infinite geometric series converges iff $\;|r|<1]\;$ ,as already mentioned.