Elementary converge and diverge Why does the geometric series $\sum^{\infty}_{j=0}c^j$ converge when $|c|<1$, but diverge when $|c|\ge 1$? Since the geometric series is $= \frac{1}{1-c}$, which means it is undefined at $c=1$, but then why would $c>1$ make it diverge for? Isn't it just altering the sign of the limit to negative to positive?

If a series converges, the terms must converge to zero. Since $|c^j| = |c|^j$, if $|c| \geq 1$, then the terms do not converge to zero, hence the series does not converge.

If $|c|<1$, then it is easy to see that $\sum_{j=0}^n c^j = \frac{1-c^{n+1}}{1-c}$, and $\lim_n c^n = 0$, hence the series converges.