Question

Suppose that the following summation notation represents a geometric series: sum j=1 ^ n a If the ratio of this series is r, which of the following represent the sum of the series? Check all that apply

Answer 1

$\sum\limits_{j=1}^n a_j$

Since

$a_j=a_{j-1}r=a_1r^{j-1}, r\ne 1$

we assume that

$\sum\limits_{j=1}^n a_j=\sum\limits_{j=1}^n a_1r^{j-1}=a_1\sum\limits_{j=1}^n r^{j-1}$

It is known, that

$(1-r^n)=(1-r)(1+r+r^2+...+r^{n-1})$

Hence

$\sum\limits_{j=1}^n r^{j-1}=\frac{1-r^n}{1-r}$

Therefore

$\sum\limits_{j=1}^n a_j=a_1\frac{1-r^n}{1-r}$

To find the sum of the series we need to find

$\lim\limits_{n\to \infty}\sum\limits_{j=1}^n a_j=\lim\limits_{n\to \infty}a_1\frac{1-r^n}{1-r}=\left\{ \begin{matrix} \infty , & r>1 \\ \frac{a_1}{1-r}, & r<1 \end{matrix} \right.$