Calculus Examples

$\sum_{r = 0}^{7} r^{2} {(- x)}^{r}$

Step 1

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

$\sum_{r = 0}^{7} r^{2} {(- x)}^{r} = \sum_{r = 1}^{7} r^{2} {(- x)}^{r} + \sum_{r = 0}^{0} r^{2} {(- x)}^{r}$

Step 2

Evaluate $\sum_{r = 1}^{7} r^{2} {(- x)}^{r}$ .

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Step 2.1

Expand the series for each value of $r$ .

$1^{2} {(- x)}^{1} + 2^{2} {(- x)}^{2} + 3^{2} {(- x)}^{3} + \dots + 7^{2} {(- x)}^{7}$

Step 2.2

Simplify the expanded form.

$0$

Step 3

Expand the series for each value of $r$ .

$0^{2} {(- x)}^{0}$

Step 4

Replace the summations with the values found.

$0 + 0^{2} {(- x)}^{0}$

Step 5

Simplify.

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Step 5.1

Add $0$ and $0^{2} {(- x)}^{0}$ .

$0^{2} {(- x)}^{0}$

Step 5.2

Raising $0$ to any positive power yields $0$ .

$0 {(- x)}^{0}$

Step 5.3

Apply the product rule to $- x$ .

$0 ({(- 1)}^{0} x^{0})$

Step 5.4

Anything raised to $0$ is $1$ .

$0 (1 x^{0})$

Step 5.5

Multiply $x^{0}$ by $1$ .

$0 x^{0}$

Step 5.6

Anything raised to $0$ is $1$ .

$0 \cdot 1$

Step 5.7

Multiply $0$ by $1$ .

$0$