Difference between revisions of "AoPS Wiki talk:Problem of the Day/September 5, 2011"
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| − | 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== | ||
| + | The series | ||
| + | <cmath>1+\frac{1}{2}+\frac{1}{4}+\cdots</cmath> | ||
| + | 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 ![\[2(1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}+\frac{1}{64}+\frac{1}{128}+\cdots)\]](http://latex.artofproblemsolving.com/8/2/9/829bf38e0288e849f8d4b0fe988cce6f5fb00053.png)
Solution
The series
![\[1+\frac{1}{2}+\frac{1}{4}+\cdots\]](http://latex.artofproblemsolving.com/2/1/5/2157cad0cf38e290384bbdeecd3d7f8b3427b3f7.png)
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 
.
