Calculus Examples

$\sum_{k = 0}^{3} 2 k^{2} - 3$

Expand the series for each value of $k$ .

$2 \cdot 0^{2} - 3 + 2 \cdot 1^{2} - 3 + 2 \cdot 2^{2} - 3 + 2 \cdot 3^{2} - 3$

Simplify.

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Raise $0$ to the power of $2$ .

$2 \cdot 0 - 3 + 2 \cdot 1^{2} - 3 + 2 \cdot 2^{2} - 3 + 2 \cdot 3^{2} - 3$

Multiply $2$ by $0$ .

$0 - 3 + 2 \cdot 1^{2} - 3 + 2 \cdot 2^{2} - 3 + 2 \cdot 3^{2} - 3$

Subtract $3$ from $0$ .

$- 3 + 2 \cdot 1^{2} - 3 + 2 \cdot 2^{2} - 3 + 2 \cdot 3^{2} - 3$

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

$- 3 + 2 \cdot 1 - 3 + 2 \cdot 2^{2} - 3 + 2 \cdot 3^{2} - 3$

Multiply $2$ by $1$ .

$- 3 + 2 - 3 + 2 \cdot 2^{2} - 3 + 2 \cdot 3^{2} - 3$

Subtract $3$ from $2$ .

$- 3 - 1 + 2 \cdot 2^{2} - 3 + 2 \cdot 3^{2} - 3$

Subtract $1$ from $- 3$ .

$- 4 + 2 \cdot 2^{2} - 3 + 2 \cdot 3^{2} - 3$

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

$- 4 + 2 \cdot 4 - 3 + 2 \cdot 3^{2} - 3$

Multiply $2$ by $4$ .

$- 4 + 8 - 3 + 2 \cdot 3^{2} - 3$

Subtract $3$ from $8$ .

$- 4 + 5 + 2 \cdot 3^{2} - 3$

Add $- 4$ and $5$ .

$1 + 2 \cdot 3^{2} - 3$

Raise $3$ to the power of $2$ .

$1 + 2 \cdot 9 - 3$

Multiply $2$ by $9$ .

$1 + 18 - 3$

Subtract $3$ from $18$ .

$1 + 15$

Add $1$ and $15$ .

$16$