Find the infinite sum, if it exists: (Help pls)

Question

Find the infinite sum, if it exists: − 2 + ( 0.5 ) + ( − 0.125 ) + ⋯

Answer 1

For an infinite geometric progression that converges (which this one does), the sum is

S = a₁ / (1-r)

where

a₁ is the first term of the geometric progression

r is the common ratio

Notice that we need to pull the leading -2 out in front, as it otherwise breaks the progression.

That leaves a₁ = 1/2 and r = -1/4:

S = -2 + (1/2) (1 / (1-(-1/4)))

= -(10/5) + (2/5)

= -8/5