Calculus Examples

$\sum_{k = 1}^{3} {(- 1)}^{k + 3} \cos (\frac{π}{k})$

Step 1

Expand the series for each value of $k$ .

${(- 1)}^{1 + 3} \cos (\frac{π}{1}) + {(- 1)}^{2 + 3} \cos (\frac{π}{2}) + {(- 1)}^{3 + 3} \cos (\frac{π}{3})$

Step 2

Simplify.

Tap for more steps...

Step 2.1

Simplify each term.

Tap for more steps...

Step 2.1.1

Add $1$ and $3$ .

${(- 1)}^{4} \cos (\frac{π}{1}) + {(- 1)}^{2 + 3} \cos (\frac{π}{2}) + {(- 1)}^{3 + 3} \cos (\frac{π}{3})$

Step 2.1.2

Raise $- 1$ to the power of $4$ .

$1 \cos (\frac{π}{1}) + {(- 1)}^{2 + 3} \cos (\frac{π}{2}) + {(- 1)}^{3 + 3} \cos (\frac{π}{3})$

Step 2.1.3

Multiply $\cos (\frac{π}{1})$ by $1$ .

$\cos (\frac{π}{1}) + {(- 1)}^{2 + 3} \cos (\frac{π}{2}) + {(- 1)}^{3 + 3} \cos (\frac{π}{3})$

Step 2.1.4

Divide $π$ by $1$ .

$\cos (π) + {(- 1)}^{2 + 3} \cos (\frac{π}{2}) + {(- 1)}^{3 + 3} \cos (\frac{π}{3})$

Step 2.1.5

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

$- \cos (0) + {(- 1)}^{2 + 3} \cos (\frac{π}{2}) + {(- 1)}^{3 + 3} \cos (\frac{π}{3})$

Step 2.1.6

The exact value of $\cos (0)$ is $1$ .

$- 1 \cdot 1 + {(- 1)}^{2 + 3} \cos (\frac{π}{2}) + {(- 1)}^{3 + 3} \cos (\frac{π}{3})$

Step 2.1.7

Multiply $- 1$ by $1$ .

$- 1 + {(- 1)}^{2 + 3} \cos (\frac{π}{2}) + {(- 1)}^{3 + 3} \cos (\frac{π}{3})$

Step 2.1.8

Add $2$ and $3$ .

$- 1 + {(- 1)}^{5} \cos (\frac{π}{2}) + {(- 1)}^{3 + 3} \cos (\frac{π}{3})$

Step 2.1.9

Raise $- 1$ to the power of $5$ .

$- 1 - 1 \cos (\frac{π}{2}) + {(- 1)}^{3 + 3} \cos (\frac{π}{3})$

Step 2.1.10

The exact value of $\cos (\frac{π}{2})$ is $0$ .

$- 1 - 1 \cdot 0 + {(- 1)}^{3 + 3} \cos (\frac{π}{3})$

Step 2.1.11

Multiply $- 1$ by $0$ .

$- 1 + 0 + {(- 1)}^{3 + 3} \cos (\frac{π}{3})$

Step 2.1.12

Add $3$ and $3$ .

$- 1 + 0 + {(- 1)}^{6} \cos (\frac{π}{3})$

Step 2.1.13

Raise $- 1$ to the power of $6$ .

$- 1 + 0 + 1 \cos (\frac{π}{3})$

Step 2.1.14

Multiply $\cos (\frac{π}{3})$ by $1$ .

$- 1 + 0 + \cos (\frac{π}{3})$

Step 2.1.15

The exact value of $\cos (\frac{π}{3})$ is $\frac{1}{2}$ .

$- 1 + 0 + \frac{1}{2}$

Step 2.2

Add $- 1$ and $0$ .

$- 1 + \frac{1}{2}$

Step 2.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- 1 \cdot \frac{2}{2} + \frac{1}{2}$

Step 2.4

Combine $- 1$ and $\frac{2}{2}$ .

$\frac{- 1 \cdot 2}{2} + \frac{1}{2}$

Step 2.5

Combine the numerators over the common denominator.

$\frac{- 1 \cdot 2 + 1}{2}$

Step 2.6

Simplify the numerator.

Tap for more steps...

Step 2.6.1

Multiply $- 1$ by $2$ .

$\frac{- 2 + 1}{2}$

Step 2.6.2

Add $- 2$ and $1$ .

$\frac{- 1}{2}$

Step 2.7

Move the negative in front of the fraction.

$- \frac{1}{2}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$- \frac{1}{2}$

Decimal Form:

$- 0.5$