Evaluate the Summation

$\sum_{x = 3}^{5} \frac{- 2}{x - 2}$

Step 1

Simplify the summation.

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Step 1.1

Move the negative in front of the fraction.

$- \frac{2}{x - 2}$

Step 1.2

Rewrite the summation.

$\sum_{x = 3}^{5} - \frac{2}{x - 2}$

Step 2

Expand the series for each value of $x$ .

$- \frac{2}{3 - 2} - \frac{2}{4 - 2} - \frac{2}{5 - 2}$

Step 3

Simplify.

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Step 3.1

Simplify each term.

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Step 3.1.1

Subtract $2$ from $3$ .

$- \frac{2}{1} - \frac{2}{4 - 2} - \frac{2}{5 - 2}$

Step 3.1.2

Divide $2$ by $1$ .

$- 1 \cdot 2 - \frac{2}{4 - 2} - \frac{2}{5 - 2}$

Step 3.1.3

Multiply $- 1$ by $2$ .

$- 2 - \frac{2}{4 - 2} - \frac{2}{5 - 2}$

Step 3.1.4

Cancel the common factor of $2$ and $4 - 2$ .

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Step 3.1.4.1

Factor $2$ out of $2$ .

$- 2 - \frac{2 (1)}{4 - 2} - \frac{2}{5 - 2}$

Step 3.1.4.2

Cancel the common factors.

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Step 3.1.4.2.1

Factor $2$ out of $4$ .

$- 2 - \frac{2 \cdot 1}{2 \cdot 2 - 2} - \frac{2}{5 - 2}$

Step 3.1.4.2.2

Factor $2$ out of $- 2$ .

$- 2 - \frac{2 (1)}{2 \cdot 2 + 2 \cdot - 1} - \frac{2}{5 - 2}$

Step 3.1.4.2.3

Factor $2$ out of $2 \cdot 2 + 2 \cdot - 1$ .

$- 2 - \frac{2 (1)}{2 \cdot (2 - 1)} - \frac{2}{5 - 2}$

Step 3.1.4.2.4

Cancel the common factor.

$- 2 - \frac{2 \cdot 1}{2 \cdot (2 - 1)} - \frac{2}{5 - 2}$

Step 3.1.4.2.5

Rewrite the expression.

$- 2 - \frac{1}{2 - 1} - \frac{2}{5 - 2}$

Step 3.1.5

Subtract $1$ from $2$ .

$- 2 - \frac{1}{1} - \frac{2}{5 - 2}$

Step 3.1.6

Cancel the common factor of $1$ .

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Step 3.1.6.1

Cancel the common factor.

$- 2 - \frac{1}{1} - \frac{2}{5 - 2}$

Step 3.1.6.2

Rewrite the expression.

$- 2 - 1 \cdot 1 - \frac{2}{5 - 2}$

Step 3.1.7

Multiply $- 1$ by $1$ .

$- 2 - 1 - \frac{2}{5 - 2}$

Step 3.1.8

Subtract $2$ from $5$ .

$- 2 - 1 - \frac{2}{3}$

Step 3.2

Find the common denominator.

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Step 3.2.1

Write $- 2$ as a fraction with denominator $1$ .

$\frac{- 2}{1} - 1 - \frac{2}{3}$

Step 3.2.2

Multiply $\frac{- 2}{1}$ by $\frac{3}{3}$ .

$\frac{- 2}{1} \cdot \frac{3}{3} - 1 - \frac{2}{3}$

Step 3.2.3

Multiply $\frac{- 2}{1}$ by $\frac{3}{3}$ .

$\frac{- 2 \cdot 3}{3} - 1 - \frac{2}{3}$

Step 3.2.4

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

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

Step 3.2.5

Multiply $\frac{- 1}{1}$ by $\frac{3}{3}$ .

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

Step 3.2.6

Multiply $\frac{- 1}{1}$ by $\frac{3}{3}$ .

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

Step 3.3

Combine the numerators over the common denominator.

$\frac{- 2 \cdot 3 - 1 \cdot 3 - 2}{3}$

Step 3.4

Simplify each term.

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Step 3.4.1

Multiply $- 2$ by $3$ .

$\frac{- 6 - 1 \cdot 3 - 2}{3}$

Step 3.4.2

Multiply $- 1$ by $3$ .

$\frac{- 6 - 3 - 2}{3}$

Step 3.5

Subtract $3$ from $- 6$ .

$\frac{- 9 - 2}{3}$

Step 3.6

Subtract $2$ from $- 9$ .

$\frac{- 11}{3}$

Step 3.7

Move the negative in front of the fraction.

$- \frac{11}{3}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$- \frac{11}{3}$

Decimal Form:

$- 3.\overline{6}$

Mixed Number Form:

$- 3 \frac{2}{3}$

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