Calculus Examples

$\sum_{k = 5}^{12} {(- 1)}^{k} \cdot 2 k$

Step 1

Split the summation to make the starting value of $k$ equal to $1$ .

$\sum_{k = 5}^{12} {(- 1)}^{k} \cdot (2 k) = \sum_{k = 1}^{12} {(- 1)}^{k} \cdot (2 k) - \sum_{k = 1}^{4} {(- 1)}^{k} \cdot (2 k)$

Step 2

Evaluate $\sum_{k = 1}^{12} {(- 1)}^{k} \cdot (2 k)$ .

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Step 2.1

Factor $2$ out of the summation.

$(2) \sum_{k = 1}^{12} {(- 1)}^{k} k$

Step 2.2

Expand the series for each value of $k$ .

${(- 1)}^{1} \cdot (2 \cdot 1) + {(- 1)}^{2} \cdot (2 \cdot 2) + {(- 1)}^{3} \cdot (2 \cdot 3) + \dots + {(- 1)}^{12} \cdot (2 \cdot 12)$

Step 2.3

Simplify the expanded form.

$12$

Step 3

Evaluate $\sum_{k = 1}^{4} {(- 1)}^{k} \cdot (2 k)$ .

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Step 3.1

Expand the series for each value of $k$ .

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

Step 3.2

Simplify.

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Step 3.2.1

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

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

Step 3.2.2

Multiply $2$ by $1$ .

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

Step 3.2.3

Multiply $- 1$ by $2$ .

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

Step 3.2.4

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

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

Step 3.2.5

Multiply $2$ by $2$ .

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

Step 3.2.6

Multiply $1$ by $4$ .

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

Step 3.2.7

Add $- 2$ and $4$ .

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

Step 3.2.8

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

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

Step 3.2.9

Multiply $2$ by $3$ .

$2 - 1 \cdot 6 + {(- 1)}^{4} \cdot (2 \cdot 4)$

Step 3.2.10

Multiply $- 1$ by $6$ .

$2 - 6 + {(- 1)}^{4} \cdot (2 \cdot 4)$

Step 3.2.11

Subtract $6$ from $2$ .

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

Step 3.2.12

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

$- 4 + 1 \cdot (2 \cdot 4)$

Step 3.2.13

Multiply $2$ by $4$ .

$- 4 + 1 \cdot 8$

Step 3.2.14

Multiply $1$ by $8$ .

$- 4 + 8$

Step 3.2.15

Add $- 4$ and $8$ .

$4$

Step 4

Replace the summations with the values found.

$12 - 4$

Step 5

Subtract $4$ from $12$ .

$8$