Calculus Examples

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

Step 1

Expand the series for each value of $k$ .

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

Step 2

Simplify.

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Step 2.1

Simplify each term.

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Step 2.1.1

Add $1$ and $1$ .

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

Step 2.1.2

Raise $- 1$ to the power of $2$ .

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

Step 2.1.3

Divide $π$ by $1$ .

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

Step 2.1.4

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.

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

Step 2.1.5

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

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

Step 2.1.6

Multiply $- 1$ by $1$ .

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

Step 2.1.7

Add $2$ and $1$ .

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

Step 2.1.8

Raise $- 1$ to the power of $3$ .

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

Step 2.1.9

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

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

Step 2.1.10

Add $3$ and $1$ .

$1 - 1 - 1 + 0 + {(- 1)}^{4} + \cos (\frac{π}{3})$

Step 2.1.11

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

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

Step 2.1.12

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

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

Step 2.2

Subtract $1$ from $1$ .

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

Step 2.3

Subtract $1$ from $0$ .

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

Step 2.4

Add $- 1$ and $0$ .

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

Step 2.5

Find the common denominator.

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Step 2.5.1

Write $- 1$ as a fraction with denominator $1$ .

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

Step 2.5.2

Multiply $\frac{- 1}{1}$ by $\frac{2}{2}$ .

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

Step 2.5.3

Multiply $\frac{- 1}{1}$ by $\frac{2}{2}$ .

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

Step 2.5.4

Write $1$ as a fraction with denominator $1$ .

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

Step 2.5.5

Multiply $\frac{1}{1}$ by $\frac{2}{2}$ .

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

Step 2.5.6

Multiply $\frac{1}{1}$ by $\frac{2}{2}$ .

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

Step 2.6

Combine the numerators over the common denominator.

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

Step 2.7

Multiply $- 1$ by $2$ .

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

Step 2.8

Add $- 2$ and $2$ .

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

Step 2.9

Add $0$ and $1$ .

$\frac{1}{2}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{2}$

Decimal Form:

$0.5$