Solve 1/1+1/2+1/4+1/8+frac{1}{16}+frac{1}{32}+frac{1}{64}+frac{1}{128}+frac{1}{256}+frac{1}{512}+frac{1}{1024}+frac{1}{2048}

$$\frac{ 1 }{ 1 } + \frac{ 1 }{ 2 } + \frac{ 1 }{ 4 } + \frac{ 1 }{ 8 } + \frac{ 1 }{ 16 } + \frac{ 1 }{ 32 } + \frac{ 1 }{ 64 } + \frac{ 1 }{ 128 } + \frac{ 1 }{ 256 } + \frac{ 1 }{ 512 } + \frac{ 1 }{ 1024 } + \frac{ 1 }{ 2048 }$$

Evaluate

$$\frac{4095}{2048}=1.999511719$$

Solution Steps

Divide 1 by 1 to get 1.

$$1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}+\frac{1}{64}+\frac{1}{128}+\frac{1}{256}+\frac{1}{512}+\frac{1}{1024}+\frac{1}{2048}$$

Convert 1 to fraction \frac{2}{2}.

$$\frac{2}{2}+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}+\frac{1}{64}+\frac{1}{128}+\frac{1}{256}+\frac{1}{512}+\frac{1}{1024}+\frac{1}{2048}$$

Since \frac{2}{2} and \frac{1}{2} have the same denominator, add them by adding their numerators.

$$\frac{2+1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}+\frac{1}{64}+\frac{1}{128}+\frac{1}{256}+\frac{1}{512}+\frac{1}{1024}+\frac{1}{2048}$$

Add 2 and 1 to get 3.

$$\frac{3}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}+\frac{1}{64}+\frac{1}{128}+\frac{1}{256}+\frac{1}{512}+\frac{1}{1024}+\frac{1}{2048}$$

Least common multiple of 2 and 4 is 4. Convert \frac{3}{2} and \frac{1}{4} to fractions with denominator 4.

$$\frac{6}{4}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}+\frac{1}{64}+\frac{1}{128}+\frac{1}{256}+\frac{1}{512}+\frac{1}{1024}+\frac{1}{2048}$$

Since \frac{6}{4} and \frac{1}{4} have the same denominator, add them by adding their numerators.

$$\frac{6+1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}+\frac{1}{64}+\frac{1}{128}+\frac{1}{256}+\frac{1}{512}+\frac{1}{1024}+\frac{1}{2048}$$

Add 6 and 1 to get 7.

$$\frac{7}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}+\frac{1}{64}+\frac{1}{128}+\frac{1}{256}+\frac{1}{512}+\frac{1}{1024}+\frac{1}{2048}$$

Least common multiple of 4 and 8 is 8. Convert \frac{7}{4} and \frac{1}{8} to fractions with denominator 8.

$$\frac{14}{8}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}+\frac{1}{64}+\frac{1}{128}+\frac{1}{256}+\frac{1}{512}+\frac{1}{1024}+\frac{1}{2048}$$

Since \frac{14}{8} and \frac{1}{8} have the same denominator, add them by adding their numerators.

$$\frac{14+1}{8}+\frac{1}{16}+\frac{1}{32}+\frac{1}{64}+\frac{1}{128}+\frac{1}{256}+\frac{1}{512}+\frac{1}{1024}+\frac{1}{2048}$$

Add 14 and 1 to get 15.

$$\frac{15}{8}+\frac{1}{16}+\frac{1}{32}+\frac{1}{64}+\frac{1}{128}+\frac{1}{256}+\frac{1}{512}+\frac{1}{1024}+\frac{1}{2048}$$

Least common multiple of 8 and 16 is 16. Convert \frac{15}{8} and \frac{1}{16} to fractions with denominator 16.

$$\frac{30}{16}+\frac{1}{16}+\frac{1}{32}+\frac{1}{64}+\frac{1}{128}+\frac{1}{256}+\frac{1}{512}+\frac{1}{1024}+\frac{1}{2048}$$

Since \frac{30}{16} and \frac{1}{16} have the same denominator, add them by adding their numerators.

$$\frac{30+1}{16}+\frac{1}{32}+\frac{1}{64}+\frac{1}{128}+\frac{1}{256}+\frac{1}{512}+\frac{1}{1024}+\frac{1}{2048}$$

Add 30 and 1 to get 31.

$$\frac{31}{16}+\frac{1}{32}+\frac{1}{64}+\frac{1}{128}+\frac{1}{256}+\frac{1}{512}+\frac{1}{1024}+\frac{1}{2048}$$

Least common multiple of 16 and 32 is 32. Convert \frac{31}{16} and \frac{1}{32} to fractions with denominator 32.

$$\frac{62}{32}+\frac{1}{32}+\frac{1}{64}+\frac{1}{128}+\frac{1}{256}+\frac{1}{512}+\frac{1}{1024}+\frac{1}{2048}$$

Since \frac{62}{32} and \frac{1}{32} have the same denominator, add them by adding their numerators.

$$\frac{62+1}{32}+\frac{1}{64}+\frac{1}{128}+\frac{1}{256}+\frac{1}{512}+\frac{1}{1024}+\frac{1}{2048}$$

Add 62 and 1 to get 63.

$$\frac{63}{32}+\frac{1}{64}+\frac{1}{128}+\frac{1}{256}+\frac{1}{512}+\frac{1}{1024}+\frac{1}{2048}$$

Least common multiple of 32 and 64 is 64. Convert \frac{63}{32} and \frac{1}{64} to fractions with denominator 64.

$$\frac{126}{64}+\frac{1}{64}+\frac{1}{128}+\frac{1}{256}+\frac{1}{512}+\frac{1}{1024}+\frac{1}{2048}$$

Since \frac{126}{64} and \frac{1}{64} have the same denominator, add them by adding their numerators.

$$\frac{126+1}{64}+\frac{1}{128}+\frac{1}{256}+\frac{1}{512}+\frac{1}{1024}+\frac{1}{2048}$$

Add 126 and 1 to get 127.

$$\frac{127}{64}+\frac{1}{128}+\frac{1}{256}+\frac{1}{512}+\frac{1}{1024}+\frac{1}{2048}$$

Least common multiple of 64 and 128 is 128. Convert \frac{127}{64} and \frac{1}{128} to fractions with denominator 128.

$$\frac{254}{128}+\frac{1}{128}+\frac{1}{256}+\frac{1}{512}+\frac{1}{1024}+\frac{1}{2048}$$

Since \frac{254}{128} and \frac{1}{128} have the same denominator, add them by adding their numerators.

$$\frac{254+1}{128}+\frac{1}{256}+\frac{1}{512}+\frac{1}{1024}+\frac{1}{2048}$$

Add 254 and 1 to get 255.

$$\frac{255}{128}+\frac{1}{256}+\frac{1}{512}+\frac{1}{1024}+\frac{1}{2048}$$

Least common multiple of 128 and 256 is 256. Convert \frac{255}{128} and \frac{1}{256} to fractions with denominator 256.

$$\frac{510}{256}+\frac{1}{256}+\frac{1}{512}+\frac{1}{1024}+\frac{1}{2048}$$

Since \frac{510}{256} and \frac{1}{256} have the same denominator, add them by adding their numerators.

$$\frac{510+1}{256}+\frac{1}{512}+\frac{1}{1024}+\frac{1}{2048}$$

Add 510 and 1 to get 511.

$$\frac{511}{256}+\frac{1}{512}+\frac{1}{1024}+\frac{1}{2048}$$

Least common multiple of 256 and 512 is 512. Convert \frac{511}{256} and \frac{1}{512} to fractions with denominator 512.

$$\frac{1022}{512}+\frac{1}{512}+\frac{1}{1024}+\frac{1}{2048}$$

Since \frac{1022}{512} and \frac{1}{512} have the same denominator, add them by adding their numerators.

$$\frac{1022+1}{512}+\frac{1}{1024}+\frac{1}{2048}$$

Add 1022 and 1 to get 1023.

$$\frac{1023}{512}+\frac{1}{1024}+\frac{1}{2048}$$

Least common multiple of 512 and 1024 is 1024. Convert \frac{1023}{512} and \frac{1}{1024} to fractions with denominator 1024.

$$\frac{2046}{1024}+\frac{1}{1024}+\frac{1}{2048}$$

Since \frac{2046}{1024} and \frac{1}{1024} have the same denominator, add them by adding their numerators.

$$\frac{2046+1}{1024}+\frac{1}{2048}$$

Add 2046 and 1 to get 2047.

$$\frac{2047}{1024}+\frac{1}{2048}$$

Least common multiple of 1024 and 2048 is 2048. Convert \frac{2047}{1024} and \frac{1}{2048} to fractions with denominator 2048.

$$\frac{4094}{2048}+\frac{1}{2048}$$

Since \frac{4094}{2048} and \frac{1}{2048} have the same denominator, add them by adding their numerators.

$$\frac{4094+1}{2048}$$

Add 4094 and 1 to get 4095.

$$\frac{4095}{2048}$$

Factor

$$\frac{3 ^ {2} \cdot 5 \cdot 7 \cdot 13}{2 ^ {11}} = 1\frac{2047}{2048} = 1.99951171875$$