Solve 1/i+1/i^3+1/i^5+1/i^7=

$$\frac { 1 } { i } + \frac { 1 } { i ^ { 3 } } + \frac { 1 } { i ^ { 5 } } + \frac { 1 } { i ^ { 7 } } =$$

Evaluate

$$0$$

Solution Steps

Multiply both numerator and denominator of \frac{1}{i} by imaginary unit i.

$$\frac{i}{-1}+\frac{1}{i^{3}}+\frac{1}{i^{5}}+\frac{1}{i^{7}}$$

Divide i by -1 to get -i.

$$-i+\frac{1}{i^{3}}+\frac{1}{i^{5}}+\frac{1}{i^{7}}$$

Calculate i to the power of 3 and get -i.

$$-i+\frac{1}{-i}+\frac{1}{i^{5}}+\frac{1}{i^{7}}$$

Multiply both numerator and denominator of \frac{1}{-i} by imaginary unit i.

$$-i+\frac{i}{1}+\frac{1}{i^{5}}+\frac{1}{i^{7}}$$

Divide i by 1 to get i.

$$-i+i+\frac{1}{i^{5}}+\frac{1}{i^{7}}$$

Add -i and i to get 0.

$$\frac{1}{i^{5}}+\frac{1}{i^{7}}$$

Calculate i to the power of 5 and get i.

$$\frac{1}{i}+\frac{1}{i^{7}}$$

Multiply both numerator and denominator of \frac{1}{i} by imaginary unit i.

$$\frac{i}{-1}+\frac{1}{i^{7}}$$

Divide i by -1 to get -i.

$$-i+\frac{1}{i^{7}}$$

Calculate i to the power of 7 and get -i.

$$-i+\frac{1}{-i}$$

Multiply both numerator and denominator of \frac{1}{-i} by imaginary unit i.

$$-i+\frac{i}{1}$$

Divide i by 1 to get i.

$$-i+i$$

Add -i and i to get 0.

$$0$$

Real Part

$$0$$

Solution Steps

Multiply both numerator and denominator of \frac{1}{i} by imaginary unit i.

$$Re(\frac{i}{-1}+\frac{1}{i^{3}}+\frac{1}{i^{5}}+\frac{1}{i^{7}})$$

Divide i by -1 to get -i.

$$Re(-i+\frac{1}{i^{3}}+\frac{1}{i^{5}}+\frac{1}{i^{7}})$$

Calculate i to the power of 3 and get -i.

$$Re(-i+\frac{1}{-i}+\frac{1}{i^{5}}+\frac{1}{i^{7}})$$

Multiply both numerator and denominator of \frac{1}{-i} by imaginary unit i.

$$Re(-i+\frac{i}{1}+\frac{1}{i^{5}}+\frac{1}{i^{7}})$$

Divide i by 1 to get i.

$$Re(-i+i+\frac{1}{i^{5}}+\frac{1}{i^{7}})$$

Add -i and i to get 0.

$$Re(\frac{1}{i^{5}}+\frac{1}{i^{7}})$$

Calculate i to the power of 5 and get i.

$$Re(\frac{1}{i}+\frac{1}{i^{7}})$$

Multiply both numerator and denominator of \frac{1}{i} by imaginary unit i.

$$Re(\frac{i}{-1}+\frac{1}{i^{7}})$$

Divide i by -1 to get -i.

$$Re(-i+\frac{1}{i^{7}})$$

Calculate i to the power of 7 and get -i.

$$Re(-i+\frac{1}{-i})$$

Multiply both numerator and denominator of \frac{1}{-i} by imaginary unit i.

$$Re(-i+\frac{i}{1})$$

Divide i by 1 to get i.

$$Re(-i+i)$$

Add -i and i to get 0.

$$Re(0)$$

The real part of 0 is 0.

$$0$$