Solve 1/1^2+1/2^2+1/3^2+1/4^2

$$\frac{ 1 }{ { 1 }^{ 2 } } + \frac{ 1 }{ { 2 }^{ 2 } } + \frac{ 1 }{ { 3 }^{ 2 } } + \frac{ 1 }{ { 4 }^{ 2 } }$$

Evaluate

$$\frac{205}{144}\approx 1.423611111$$

Solution Steps

Rewrite 1^{2} as 1\times 1. Cancel out 1 in both numerator and denominator.

$$\frac{1}{1}+\frac{1}{2^{2}}+\frac{1}{3^{2}}+\frac{1}{4^{2}}$$

Divide 1 by 1 to get 1.

$$1+\frac{1}{2^{2}}+\frac{1}{3^{2}}+\frac{1}{4^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$1+\frac{1}{4}+\frac{1}{3^{2}}+\frac{1}{4^{2}}$$

Convert 1 to fraction \frac{4}{4}.

$$\frac{4}{4}+\frac{1}{4}+\frac{1}{3^{2}}+\frac{1}{4^{2}}$$

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

$$\frac{4+1}{4}+\frac{1}{3^{2}}+\frac{1}{4^{2}}$$

Add 4 and 1 to get 5.

$$\frac{5}{4}+\frac{1}{3^{2}}+\frac{1}{4^{2}}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{5}{4}+\frac{1}{9}+\frac{1}{4^{2}}$$

Least common multiple of 4 and 9 is 36. Convert \frac{5}{4} and \frac{1}{9} to fractions with denominator 36.

$$\frac{45}{36}+\frac{4}{36}+\frac{1}{4^{2}}$$

Since \frac{45}{36} and \frac{4}{36} have the same denominator, add them by adding their numerators.

$$\frac{45+4}{36}+\frac{1}{4^{2}}$$

Add 45 and 4 to get 49.

$$\frac{49}{36}+\frac{1}{4^{2}}$$

Calculate 4 to the power of 2 and get 16.

$$\frac{49}{36}+\frac{1}{16}$$

Least common multiple of 36 and 16 is 144. Convert \frac{49}{36} and \frac{1}{16} to fractions with denominator 144.

$$\frac{196}{144}+\frac{9}{144}$$

Since \frac{196}{144} and \frac{9}{144} have the same denominator, add them by adding their numerators.

$$\frac{196+9}{144}$$

Add 196 and 9 to get 205.

$$\frac{205}{144}$$

Factor

$$\frac{5 \cdot 41}{2 ^ {4} \cdot 3 ^ {2}} = 1\frac{61}{144} = 1.4236111111111112$$