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

Nov 14, 2015

This is an arithmetic series, not a geometric one.

Its sum diverges to $- \infty$.

Explanation:

The terms of this series are of the form ${a}_{n} = {a}_{1} + \left(n - 1\right) d$, with initial term ${a}_{1} = 4$ and common difference $d = - 2$

${\sum}_{n = 1}^{N} {a}_{n} = {\sum}_{n = 1}^{N} \left({a}_{1} + \left(n - 1\right) d\right) = N \left({a}_{1} + \frac{\left(N - 1\right) d}{2}\right)$

$= 4 N + \frac{N \left(N - 1\right) \cdot \left(- 2\right)}{2} = 4 N - N \left(N - 1\right)$

$= 5 N - {N}^{2} \to - \infty$ as $N \to \infty$