Calculus Examples

$\sum_{k = 2}^{5} \frac{1}{2 k (k + 1)}$

Step 1

Expand the series for each value of $k$ .

$\frac{1}{2 \cdot 2 (2 + 1)} + \frac{1}{2 \cdot 3 (3 + 1)} + \frac{1}{2 \cdot 4 (4 + 1)} + \frac{1}{2 \cdot 5 (5 + 1)}$

Step 2

Simplify.

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Step 2.1

Simplify each term.

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Step 2.1.1

Multiply $2$ by $2$ .

$\frac{1}{4 (2 + 1)} + \frac{1}{2 \cdot 3 (3 + 1)} + \frac{1}{2 \cdot 4 (4 + 1)} + \frac{1}{2 \cdot 5 (5 + 1)}$

Step 2.1.2

Add $2$ and $1$ .

$\frac{1}{4 \cdot 3} + \frac{1}{2 \cdot 3 (3 + 1)} + \frac{1}{2 \cdot 4 (4 + 1)} + \frac{1}{2 \cdot 5 (5 + 1)}$

Step 2.1.3

Multiply $4$ by $3$ .

$\frac{1}{12} + \frac{1}{2 \cdot 3 (3 + 1)} + \frac{1}{2 \cdot 4 (4 + 1)} + \frac{1}{2 \cdot 5 (5 + 1)}$

Step 2.1.4

Multiply $2$ by $3$ .

$\frac{1}{12} + \frac{1}{6 (3 + 1)} + \frac{1}{2 \cdot 4 (4 + 1)} + \frac{1}{2 \cdot 5 (5 + 1)}$

Step 2.1.5

Add $3$ and $1$ .

$\frac{1}{12} + \frac{1}{6 \cdot 4} + \frac{1}{2 \cdot 4 (4 + 1)} + \frac{1}{2 \cdot 5 (5 + 1)}$

Step 2.1.6

Multiply $6$ by $4$ .

$\frac{1}{12} + \frac{1}{24} + \frac{1}{2 \cdot 4 (4 + 1)} + \frac{1}{2 \cdot 5 (5 + 1)}$

Step 2.1.7

Multiply $2$ by $4$ .

$\frac{1}{12} + \frac{1}{24} + \frac{1}{8 (4 + 1)} + \frac{1}{2 \cdot 5 (5 + 1)}$

Step 2.1.8

Add $4$ and $1$ .

$\frac{1}{12} + \frac{1}{24} + \frac{1}{8 \cdot 5} + \frac{1}{2 \cdot 5 (5 + 1)}$

Step 2.1.9

Multiply $8$ by $5$ .

$\frac{1}{12} + \frac{1}{24} + \frac{1}{40} + \frac{1}{2 \cdot 5 (5 + 1)}$

Step 2.1.10

Multiply $2$ by $5$ .

$\frac{1}{12} + \frac{1}{24} + \frac{1}{40} + \frac{1}{10 (5 + 1)}$

Step 2.1.11

Add $5$ and $1$ .

$\frac{1}{12} + \frac{1}{24} + \frac{1}{40} + \frac{1}{10 \cdot 6}$

Step 2.1.12

Multiply $10$ by $6$ .

$\frac{1}{12} + \frac{1}{24} + \frac{1}{40} + \frac{1}{60}$

Step 2.2

Find the common denominator.

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Step 2.2.1

Multiply $\frac{1}{12}$ by $\frac{10}{10}$ .

$\frac{1}{12} \cdot \frac{10}{10} + \frac{1}{24} + \frac{1}{40} + \frac{1}{60}$

Step 2.2.2

Multiply $\frac{1}{12}$ by $\frac{10}{10}$ .

$\frac{10}{12 \cdot 10} + \frac{1}{24} + \frac{1}{40} + \frac{1}{60}$

Step 2.2.3

Multiply $\frac{1}{24}$ by $\frac{5}{5}$ .

$\frac{10}{12 \cdot 10} + \frac{1}{24} \cdot \frac{5}{5} + \frac{1}{40} + \frac{1}{60}$

Step 2.2.4

Multiply $\frac{1}{24}$ by $\frac{5}{5}$ .

$\frac{10}{12 \cdot 10} + \frac{5}{24 \cdot 5} + \frac{1}{40} + \frac{1}{60}$

Step 2.2.5

Multiply $\frac{1}{40}$ by $\frac{3}{3}$ .

$\frac{10}{12 \cdot 10} + \frac{5}{24 \cdot 5} + \frac{1}{40} \cdot \frac{3}{3} + \frac{1}{60}$

Step 2.2.6

Multiply $\frac{1}{40}$ by $\frac{3}{3}$ .

$\frac{10}{12 \cdot 10} + \frac{5}{24 \cdot 5} + \frac{3}{40 \cdot 3} + \frac{1}{60}$

Step 2.2.7

Multiply $\frac{1}{60}$ by $\frac{2}{2}$ .

$\frac{10}{12 \cdot 10} + \frac{5}{24 \cdot 5} + \frac{3}{40 \cdot 3} + \frac{1}{60} \cdot \frac{2}{2}$

Step 2.2.8

Multiply $\frac{1}{60}$ by $\frac{2}{2}$ .

$\frac{10}{12 \cdot 10} + \frac{5}{24 \cdot 5} + \frac{3}{40 \cdot 3} + \frac{2}{60 \cdot 2}$

Step 2.2.9

Reorder the factors of $12 \cdot 10$ .

$\frac{10}{10 \cdot 12} + \frac{5}{24 \cdot 5} + \frac{3}{40 \cdot 3} + \frac{2}{60 \cdot 2}$

Step 2.2.10

Multiply $10$ by $12$ .

$\frac{10}{120} + \frac{5}{24 \cdot 5} + \frac{3}{40 \cdot 3} + \frac{2}{60 \cdot 2}$

Step 2.2.11

Reorder the factors of $24 \cdot 5$ .

$\frac{10}{120} + \frac{5}{5 \cdot 24} + \frac{3}{40 \cdot 3} + \frac{2}{60 \cdot 2}$

Step 2.2.12

Multiply $5$ by $24$ .

$\frac{10}{120} + \frac{5}{120} + \frac{3}{40 \cdot 3} + \frac{2}{60 \cdot 2}$

Step 2.2.13

Reorder the factors of $40 \cdot 3$ .

$\frac{10}{120} + \frac{5}{120} + \frac{3}{3 \cdot 40} + \frac{2}{60 \cdot 2}$

Step 2.2.14

Multiply $3$ by $40$ .

$\frac{10}{120} + \frac{5}{120} + \frac{3}{120} + \frac{2}{60 \cdot 2}$

Step 2.2.15

Reorder the factors of $60 \cdot 2$ .

$\frac{10}{120} + \frac{5}{120} + \frac{3}{120} + \frac{2}{2 \cdot 60}$

Step 2.2.16

Multiply $2$ by $60$ .

$\frac{10}{120} + \frac{5}{120} + \frac{3}{120} + \frac{2}{120}$

Step 2.3

Combine the numerators over the common denominator.

$\frac{10 + 5 + 3 + 2}{120}$

Step 2.4

Add $10$ and $5$ .

$\frac{15 + 3 + 2}{120}$

Step 2.5

Add $15$ and $3$ .

$\frac{18 + 2}{120}$

Step 2.6

Add $18$ and $2$ .

$\frac{20}{120}$

Step 2.7

Cancel the common factor of $20$ and $120$ .

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Step 2.7.1

Factor $20$ out of $20$ .

$\frac{20 (1)}{120}$

Step 2.7.2

Cancel the common factors.

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Step 2.7.2.1

Factor $20$ out of $120$ .

$\frac{20 \cdot 1}{20 \cdot 6}$

Step 2.7.2.2

Cancel the common factor.

$\frac{20 \cdot 1}{20 \cdot 6}$

Step 2.7.2.3

Rewrite the expression.

$\frac{1}{6}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{6}$

Decimal Form:

$0.1\overline{6}$