Question

How do you find the sum of the infinite geometric series 1-2+4-..+1024?

Answer 1

Apply the geometric series formula to find
`1 - 2 + 4 - ... + 1024 = 683`

Explanation:

Given a geometric series with initial value `a` and common ratio `r`, we have the formula

`sum_(k=0)^(n-1)ar^k = a(1-r^n)/(1-r)`

(See What is the formula for the sum of a geometric sequence? for a proof)

In our exercise, the first term is clearly `a=1` and we can find the common ratio by dividing the second term by the first to obtain
`r = (-2)/1 = -2`

Finally, we can find that `n=11` by noting that
`1024 = (-2)^10 = r^10`

`=> 1 - 2 + 4 -... + 1024 = sum_(k=0)^(11-1)1*(-2)^k`

Thus, applying the formula gives us

`sum_(k=0)^(11-1)1*(-2)^k = 1(1-(-2)^11)/(1-(-2)) = 2049/3 = 683`