Question

Find the sum of the infinite geometric series 8 + 4 + 2 + 1 +…if it exists

Answer 1

a0=8
r=1/2
sum=a0/1-r=8/(1-1/2)=16

Answer 2

The series seem to be \[\sum_{n = -3}^{\infty} 2^{-n}\]
We know that \[\sum_{n = 0}^{\infty} r^{-n} = { 1 \over {1-1/r} }\]
So if we break up the sum we can apply this law. As such:
\[\sum_{n = -3}^{\infty} 2^{-n} = \sum_{n = 0}^{\infty} 2^{-n} + 2^1 + 2^2 + 2^3 = {1 \over 1 - 1/2} + 2 + 4 + 8 = 2 + 14 =16\]