Solve 1div2^11+1div2^12+1div2^13+1div2^14+1div{2}^15+1div{2}^16

$$1 \div { 2 }^{ 11 } +1 \div { 2 }^{ 12 } +1 \div { 2 }^{ 13 } +1 \div { 2 }^{ 14 } +1 \div { 2 }^{ 15 } +1 \div { 2 }^{ 16 }$$

Evaluate

$$\frac{63}{65536}=0.000961304$$

Solution Steps

Calculate 2 to the power of 11 and get 2048.

$$\frac{1}{2048}+\frac{1}{2^{12}}+\frac{1}{2^{13}}+\frac{1}{2^{14}}+\frac{1}{2^{15}}+\frac{1}{2^{16}}$$

Calculate 2 to the power of 12 and get 4096.

$$\frac{1}{2048}+\frac{1}{4096}+\frac{1}{2^{13}}+\frac{1}{2^{14}}+\frac{1}{2^{15}}+\frac{1}{2^{16}}$$

Least common multiple of 2048 and 4096 is 4096. Convert \frac{1}{2048} and \frac{1}{4096} to fractions with denominator 4096.

$$\frac{2}{4096}+\frac{1}{4096}+\frac{1}{2^{13}}+\frac{1}{2^{14}}+\frac{1}{2^{15}}+\frac{1}{2^{16}}$$

Since \frac{2}{4096} and \frac{1}{4096} have the same denominator, add them by adding their numerators.

$$\frac{2+1}{4096}+\frac{1}{2^{13}}+\frac{1}{2^{14}}+\frac{1}{2^{15}}+\frac{1}{2^{16}}$$

Add 2 and 1 to get 3.

$$\frac{3}{4096}+\frac{1}{2^{13}}+\frac{1}{2^{14}}+\frac{1}{2^{15}}+\frac{1}{2^{16}}$$

Calculate 2 to the power of 13 and get 8192.

$$\frac{3}{4096}+\frac{1}{8192}+\frac{1}{2^{14}}+\frac{1}{2^{15}}+\frac{1}{2^{16}}$$

Least common multiple of 4096 and 8192 is 8192. Convert \frac{3}{4096} and \frac{1}{8192} to fractions with denominator 8192.

$$\frac{6}{8192}+\frac{1}{8192}+\frac{1}{2^{14}}+\frac{1}{2^{15}}+\frac{1}{2^{16}}$$

Since \frac{6}{8192} and \frac{1}{8192} have the same denominator, add them by adding their numerators.

$$\frac{6+1}{8192}+\frac{1}{2^{14}}+\frac{1}{2^{15}}+\frac{1}{2^{16}}$$

Add 6 and 1 to get 7.

$$\frac{7}{8192}+\frac{1}{2^{14}}+\frac{1}{2^{15}}+\frac{1}{2^{16}}$$

Calculate 2 to the power of 14 and get 16384.

$$\frac{7}{8192}+\frac{1}{16384}+\frac{1}{2^{15}}+\frac{1}{2^{16}}$$

Least common multiple of 8192 and 16384 is 16384. Convert \frac{7}{8192} and \frac{1}{16384} to fractions with denominator 16384.

$$\frac{14}{16384}+\frac{1}{16384}+\frac{1}{2^{15}}+\frac{1}{2^{16}}$$

Since \frac{14}{16384} and \frac{1}{16384} have the same denominator, add them by adding their numerators.

$$\frac{14+1}{16384}+\frac{1}{2^{15}}+\frac{1}{2^{16}}$$

Add 14 and 1 to get 15.

$$\frac{15}{16384}+\frac{1}{2^{15}}+\frac{1}{2^{16}}$$

Calculate 2 to the power of 15 and get 32768.

$$\frac{15}{16384}+\frac{1}{32768}+\frac{1}{2^{16}}$$

Least common multiple of 16384 and 32768 is 32768. Convert \frac{15}{16384} and \frac{1}{32768} to fractions with denominator 32768.

$$\frac{30}{32768}+\frac{1}{32768}+\frac{1}{2^{16}}$$

Since \frac{30}{32768} and \frac{1}{32768} have the same denominator, add them by adding their numerators.

$$\frac{30+1}{32768}+\frac{1}{2^{16}}$$

Add 30 and 1 to get 31.

$$\frac{31}{32768}+\frac{1}{2^{16}}$$

Calculate 2 to the power of 16 and get 65536.

$$\frac{31}{32768}+\frac{1}{65536}$$

Least common multiple of 32768 and 65536 is 65536. Convert \frac{31}{32768} and \frac{1}{65536} to fractions with denominator 65536.

$$\frac{62}{65536}+\frac{1}{65536}$$

Since \frac{62}{65536} and \frac{1}{65536} have the same denominator, add them by adding their numerators.

$$\frac{62+1}{65536}$$

Add 62 and 1 to get 63.

$$\frac{63}{65536}$$

Factor

$$\frac{3 ^ {2} \cdot 7}{2 ^ {16}} = 0.0009613037109375$$