Calculus Examples

$3 \sum_{k = 3}^{6} \frac{1}{k - 1}$

Step 1

Expand the series for each value of $k$ .

$3 (\frac{1}{3 - 1} + \frac{1}{4 - 1} + \frac{1}{5 - 1} + \frac{1}{6 - 1})$

Step 2

Simplify.

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Step 2.1

Simplify each term.

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Step 2.1.1

Subtract $1$ from $3$ .

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

Step 2.1.2

Subtract $1$ from $4$ .

$3 (\frac{1}{2} + \frac{1}{3} + \frac{1}{5 - 1} + \frac{1}{6 - 1})$

Step 2.1.3

Subtract $1$ from $5$ .

$3 (\frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{6 - 1})$

Step 2.1.4

Subtract $1$ from $6$ .

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

Step 2.2

Find the common denominator.

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Step 2.2.1

Multiply $\frac{1}{2}$ by $\frac{30}{30}$ .

$3 (\frac{1}{2} \cdot \frac{30}{30} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5})$

Step 2.2.2

Multiply $\frac{1}{2}$ by $\frac{30}{30}$ .

$3 (\frac{30}{2 \cdot 30} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5})$

Step 2.2.3

Multiply $\frac{1}{3}$ by $\frac{20}{20}$ .

$3 (\frac{30}{2 \cdot 30} + \frac{1}{3} \cdot \frac{20}{20} + \frac{1}{4} + \frac{1}{5})$

Step 2.2.4

Multiply $\frac{1}{3}$ by $\frac{20}{20}$ .

$3 (\frac{30}{2 \cdot 30} + \frac{20}{3 \cdot 20} + \frac{1}{4} + \frac{1}{5})$

Step 2.2.5

Multiply $\frac{1}{4}$ by $\frac{15}{15}$ .

$3 (\frac{30}{2 \cdot 30} + \frac{20}{3 \cdot 20} + \frac{1}{4} \cdot \frac{15}{15} + \frac{1}{5})$

Step 2.2.6

Multiply $\frac{1}{4}$ by $\frac{15}{15}$ .

$3 (\frac{30}{2 \cdot 30} + \frac{20}{3 \cdot 20} + \frac{15}{4 \cdot 15} + \frac{1}{5})$

Step 2.2.7

Multiply $\frac{1}{5}$ by $\frac{12}{12}$ .

$3 (\frac{30}{2 \cdot 30} + \frac{20}{3 \cdot 20} + \frac{15}{4 \cdot 15} + \frac{1}{5} \cdot \frac{12}{12})$

Step 2.2.8

Multiply $\frac{1}{5}$ by $\frac{12}{12}$ .

$3 (\frac{30}{2 \cdot 30} + \frac{20}{3 \cdot 20} + \frac{15}{4 \cdot 15} + \frac{12}{5 \cdot 12})$

Step 2.2.9

Multiply $2$ by $30$ .

$3 (\frac{30}{60} + \frac{20}{3 \cdot 20} + \frac{15}{4 \cdot 15} + \frac{12}{5 \cdot 12})$

Step 2.2.10

Multiply $3$ by $20$ .

$3 (\frac{30}{60} + \frac{20}{60} + \frac{15}{4 \cdot 15} + \frac{12}{5 \cdot 12})$

Step 2.2.11

Multiply $4$ by $15$ .

$3 (\frac{30}{60} + \frac{20}{60} + \frac{15}{60} + \frac{12}{5 \cdot 12})$

Step 2.2.12

Multiply $5$ by $12$ .

$3 (\frac{30}{60} + \frac{20}{60} + \frac{15}{60} + \frac{12}{60})$

Step 2.3

Combine the numerators over the common denominator.

$3 \frac{30 + 20 + 15 + 12}{60}$

Step 2.4

Add $30$ and $20$ .

$3 \frac{50 + 15 + 12}{60}$

Step 2.5

Add $50$ and $15$ .

$3 \frac{65 + 12}{60}$

Step 2.6

Add $65$ and $12$ .

$3 (\frac{77}{60})$

Step 2.7

Cancel the common factor of $3$ .

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Step 2.7.1

Factor $3$ out of $60$ .

$3 \frac{77}{3 (20)}$

Step 2.7.2

Cancel the common factor.

$3 \frac{77}{3 \cdot 20}$

Step 2.7.3

Rewrite the expression.

$\frac{77}{20}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\frac{77}{20}$

Decimal Form:

$3.85$

Mixed Number Form:

$3 \frac{17}{20}$