Question

How do you find the sum of the infinite geometric series 1 + 0.4 + 0.16 + 0.064 + . . .?

Answer 1

`5/3`

Explanation:

The terms you're summing are `a_n=(4/10)^n`. In fact,

`a_0 = (4/10)^0 = 1`

`a_1 = (4/10)^1=4/10=0.4`

`a_2= (4/10)^2 = 16/100=0.16`, and so on.

The general rule states that, if a series of the form

`sum_{n=0}^infty k^n`

converges, then in converges to `\frac{1}{1-k}`

Since in your case `k=4/10`, the result is

`1/(1-4/10) = 1/(1-2/5) = 1/(3/5) = 5/3`