Difference between revisions of "AoPS Wiki talk:Problem of the Day/September 5, 2011"
(3 intermediate revisions by 3 users not shown) | |||
Line 1: | Line 1: | ||
− | 4 | + | ==Problem== |
+ | Simplify <cmath> 2(1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}+\frac{1}{64}+\frac{1}{128}+\cdots) </cmath> | ||
+ | |||
==Solution== | ==Solution== | ||
The series | The series | ||
<cmath>1+\frac{1}{2}+\frac{1}{4}+\cdots</cmath> | <cmath>1+\frac{1}{2}+\frac{1}{4}+\cdots</cmath> | ||
− | is a geometric series, which famously converges to 2. Thus, since the problem asks for twice this number, the answer is <math>\boxed{4}</math>. | + | is a geometric series, which famously converges to 2. This can by applying the formula for infinite geometric sums, <math> \frac{a_1}{1-r} </math> where <math> a_1 </math> is the first term and <math> r </math> is the ratio. For this series, the sum is <math> \frac{1}{1-\frac{1}{2}}=2 </math>. Thus, since the problem asks for twice this number, the answer is <math>\boxed{4}</math>. |
Latest revision as of 08:28, 7 September 2011
Problem
Simplify
Solution
The series is a geometric series, which famously converges to 2. This can by applying the formula for infinite geometric sums, where is the first term and is the ratio. For this series, the sum is . Thus, since the problem asks for twice this number, the answer is .