Question

Find the summation. I know how to do these in general but I have never seen one with infinity. Please explain how to do this. Thank you.

Answer 1

`\bf \textit{sum of an infinite geometric series}\\\\ S=\sum\limits_(i=0)^(\infty)\ a_1r^i\implies \cfrac{a_1}{1-r}\qquad \begin{cases} a_1=\textit{first term}\\ r=\textit{common ratio} \end{cases}`


`\bf -----------------------------\\\\ \sum\limits_(n=1)^(\infty)\ 1400(-0.5)^(n-1)\qquad \begin{cases} a_1=1400\\ r=-0.5\to -(1)/(2) \end{cases} \\\\\\ \cfrac{1400}{1-\left(-(1)/(2) \right)}\implies \cfrac{1400}{1+(1)/(2)}\implies \cfrac{1400}{(3)/(2)}\implies \cfrac{1400}{1}\cdot \cfrac{2}{3}\implies \cfrac{2800}{3}`