Calculus Examples

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

Step 1

Expand the series for each value of $k$ .

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

Step 2

Simplify.

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Step 2.1

Add $1$ and $3$ .

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

Step 2.2

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

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

Step 2.3

Evaluate $\cos (1 π)$ .

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

Step 2.4

Multiply $1$ by $- 1$ .

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

Step 2.5

Add $2$ and $3$ .

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

Step 2.6

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

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

Step 2.7

Evaluate $\cos (2 π)$ .

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

Step 2.8

Multiply $- 1$ by $1$ .

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

Step 2.9

Subtract $1$ from $- 1$ .

$- 2 + {(- 1)}^{3 + 3} \cos (3 π)$

Step 2.10

Add $3$ and $3$ .

$- 2 + {(- 1)}^{6} \cos (3 π)$

Step 2.11

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

$- 2 + 1 \cos (3 π)$

Step 2.12

Evaluate $\cos (3 π)$ .

$- 2 + 1 \cdot - 1$

Step 2.13

Multiply $1$ by $- 1$ .

$- 2 - 1$

Step 2.14

Subtract $1$ from $- 2$ .

$- 3$