Question

How do you find the sum of the infinite geometric series 4+2+0-2...?

Answer 1

This is an arithmetic series, not a geometric one.

Its sum diverges to `-oo`.

Explanation:

The terms of this series are of the form `a_n = a_1 + (n-1)d`, with initial term `a_1 = 4` and common difference `d = -2`

`sum_(n=1)^N a_n = sum_(n=1)^N (a_1 + (n-1)d) = N(a_1 + ((N-1)d)/2)`

`=4N + (N(N-1)*(-2))/2 = 4N-N(N-1)`

`= 5N-N^2 -> -oo` as `N->oo`