Calculus Examples

$\sum_{1}^{8} 1 - \frac{11}{x} + x^{2}$

Step 1

Split the summation into smaller summations that fit the summation rules.

$\sum_{1}^{8} 1 - \frac{11}{x} + x^{2} = \sum_{x = 1}^{8} 1 + \sum_{x = 1}^{8} - \frac{11}{x} + \sum_{x = 1}^{8} x^{2}$

Step 2

Evaluate $\sum_{x = 1}^{8} 1$ .

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Step 2.1

The formula for the summation of a constant is:

$\sum_{k = 1}^{n} c = c n$

Step 2.2

Substitute the values into the formula.

$(1) (8)$

Step 2.3

Multiply $8$ by $1$ .

$8$

Step 3

Evaluate $\sum_{x = 1}^{8} - \frac{11}{x}$ .

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Step 3.1

Factor $- 1$ out of the summation.

$(- 1) \sum_{x = 1}^{8} \frac{11}{x}$

Step 3.2

Expand the series for each value of $x$ .

$- \frac{11}{1} - \frac{11}{2} - \frac{11}{3} + \dots - \frac{11}{8}$

Step 3.3

Simplify the expanded form.

$- \frac{8371}{280}$

Step 4

Evaluate $\sum_{x = 1}^{8} x^{2}$ .

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Step 4.1

The formula for the summation of a polynomial with degree $2$ is:

$\sum_{k = 1}^{n} k^{2} = \frac{n (n + 1) (2 n + 1)}{6}$

Step 4.2

Substitute the values into the formula.

$\frac{8 (8 + 1) (2 \cdot 8 + 1)}{6}$

Step 4.3

Simplify.

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Step 4.3.1

Cancel the common factor of $8$ and $6$ .

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Step 4.3.1.1

Factor $2$ out of $8 (8 + 1) (2 \cdot 8 + 1)$ .

$\frac{2 (4 (8 + 1) (2 \cdot 8 + 1))}{6}$

Step 4.3.1.2

Cancel the common factors.

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Step 4.3.1.2.1

Factor $2$ out of $6$ .

$\frac{2 (4 (8 + 1) (2 \cdot 8 + 1))}{2 (3)}$

Step 4.3.1.2.2

Cancel the common factor.

$\frac{2 (4 (8 + 1) (2 \cdot 8 + 1))}{2 \cdot 3}$

Step 4.3.1.2.3

Rewrite the expression.

$\frac{4 (8 + 1) (2 \cdot 8 + 1)}{3}$

Step 4.3.2

Simplify the numerator.

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Step 4.3.2.1

Multiply $2$ by $8$ .

$\frac{4 (8 + 1) (16 + 1)}{3}$

Step 4.3.2.2

Add $8$ and $1$ .

$\frac{4 \cdot 9 (16 + 1)}{3}$

Step 4.3.2.3

Multiply $4$ by $9$ .

$\frac{36 (16 + 1)}{3}$

Step 4.3.2.4

Add $16$ and $1$ .

$\frac{36 \cdot 17}{3}$

Step 4.3.3

Simplify the expression.

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Step 4.3.3.1

Multiply $36$ by $17$ .

$\frac{612}{3}$

Step 4.3.3.2

Divide $612$ by $3$ .

$204$

Step 5

Add the results of the summations.

$8 - \frac{8371}{280} + 204$

Step 6

Simplify.

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Step 6.1

Find the common denominator.

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Step 6.1.1

Write $8$ as a fraction with denominator $1$ .

$\frac{8}{1} - \frac{8371}{280} + 204$

Step 6.1.2

Multiply $\frac{8}{1}$ by $\frac{280}{280}$ .

$\frac{8}{1} \cdot \frac{280}{280} - \frac{8371}{280} + 204$

Step 6.1.3

Multiply $\frac{8}{1}$ by $\frac{280}{280}$ .

$\frac{8 \cdot 280}{280} - \frac{8371}{280} + 204$

Step 6.1.4

Write $204$ as a fraction with denominator $1$ .

$\frac{8 \cdot 280}{280} - \frac{8371}{280} + \frac{204}{1}$

Step 6.1.5

Multiply $\frac{204}{1}$ by $\frac{280}{280}$ .

$\frac{8 \cdot 280}{280} - \frac{8371}{280} + \frac{204}{1} \cdot \frac{280}{280}$

Step 6.1.6

Multiply $\frac{204}{1}$ by $\frac{280}{280}$ .

$\frac{8 \cdot 280}{280} - \frac{8371}{280} + \frac{204 \cdot 280}{280}$

Step 6.2

Combine the numerators over the common denominator.

$\frac{8 \cdot 280 - 8371 + 204 \cdot 280}{280}$

Step 6.3

Simplify each term.

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Step 6.3.1

Multiply $8$ by $280$ .

$\frac{2240 - 8371 + 204 \cdot 280}{280}$

Step 6.3.2

Multiply $204$ by $280$ .

$\frac{2240 - 8371 + 57120}{280}$

Step 6.4

Simplify by adding and subtracting.

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Step 6.4.1

Subtract $8371$ from $2240$ .

$\frac{- 6131 + 57120}{280}$

Step 6.4.2

Add $- 6131$ and $57120$ .

$\frac{50989}{280}$