Calculus Examples

$\sum_{k = 1}^{4} {(- 1)}^{k} \cos (k π)$

Step 1

Expand the series for each value of $k$ .

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

Step 2

Simplify.

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Step 2.1

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

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

Step 2.2

Evaluate $\cos (1 π)$ .

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

Step 2.3

Multiply $- 1$ by $- 1$ .

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

Step 2.4

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

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

Step 2.5

Evaluate $\cos (2 π)$ .

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

Step 2.6

Multiply $1$ by $1$ .

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

Step 2.7

Add $1$ and $1$ .

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

Step 2.8

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

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

Step 2.9

Evaluate $\cos (3 π)$ .

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

Step 2.10

Multiply $- 1$ by $- 1$ .

$2 + 1 + {(- 1)}^{4} \cos (4 π)$

Step 2.11

Add $2$ and $1$ .

$3 + {(- 1)}^{4} \cos (4 π)$

Step 2.12

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

$3 + 1 \cos (4 π)$

Step 2.13

Evaluate $\cos (4 π)$ .

$3 + 1 \cdot 1$

Step 2.14

Multiply $1$ by $1$ .

$3 + 1$

Step 2.15

Add $3$ and $1$ .

$4$