The sequence converges, since each term is -1/4 times the previous term, and abs(-1/4) < 1. Thus the infinite sum exists. Let us call this sum S. Then:
color(white)(–1/2)S=-2+1/2-1/8+...
color(white)(–1/2S)=-2(1-1/4+1/16-1/64+...)
color(blue)(–1/2 S=" "1-1/4+1/16-1/64+...)
If we multiply both sides by -1/4, we get
(–1/4)(–1/2 S)=(–1/4)(1-1/4+1/16-1/64+...)
color(green)(" "1/8 S = -1/4 + 1/16 - 1/64 + ...)
Notice how the expansion for color(blue)(-1/2 S) includes the expansion for color(green)(1/8 S). If we subtract 1/8 S from -1/2 S we get:
" "color(blue)(–1/2 S) - color(green)(1/8 S)=color(blue)(" "1-1/4+1/16-1/64+...)
color(white)(–1/2 S - 1/8 S=)color(green)(" "-(-1/4+1/16-1/64+...))
" "–5/8 S=1
Solving for S gives:
S=-8/5