AoPS Wiki talk:Problem of the Day/September 5, 2011

Problem

Simplify

\[2(1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\fac{1}{32}+\frac{1}{64}+\frac{1}{128}+\cdots)\] (Error compiling LaTeX. Unknown error_msg)

Solution

The series $1+\frac{1}{2}+\frac{1}{4}+\cdots$ is a geometric series, which famously converges to 2. This can by applying the formula for infinite geometric sums, $\frac{a_1}{1-r}$ where $a_1$ is the first term and $r$ is the ratio. For this series, the sum is $\frac{1}{1-\frac{1}{2}}=2$ . Thus, since the problem asks for twice this number, the answer is $\boxed{4}$ .

Page

Toolbox

Search

AoPS Wiki talk:Problem of the Day/September 5, 2011

Problem

Solution