Monthly Archives: August 2013

Limit: Geometric Sequence

If \( a_n = r^n \) is an infinite geometric sequence of real numbers and \( 0 < r < 1 \) then

$$ \lim_{ n \rightarrow \infty } r^n = 0 $$

Without loss of generality \( r \) can be defined as \( r = \frac{1}{ 1 + c } \) for \( c \gt 0 \).

Using Bernoulli’s Inequality

$$ r^n = {\left( \frac{1}{1 + c} \right)}^n = \frac{1}{\left( 1 + c \right)^n} \le \frac{1}{1 + nc} \le \frac{1}{nc} $$