Calculus Examples

$\sum_{j = 3}^{5} \frac{1}{j}$

Step 1

Expand the series for each value of $j$ .

$\frac{1}{3} + \frac{1}{4} + \frac{1}{5}$

Step 2

Simplify.

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Step 2.1

Find the common denominator.

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Step 2.1.1

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

$\frac{1}{3} \cdot \frac{20}{20} + \frac{1}{4} + \frac{1}{5}$

Step 2.1.2

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

$\frac{20}{3 \cdot 20} + \frac{1}{4} + \frac{1}{5}$

Step 2.1.3

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

$\frac{20}{3 \cdot 20} + \frac{1}{4} \cdot \frac{15}{15} + \frac{1}{5}$

Step 2.1.4

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

$\frac{20}{3 \cdot 20} + \frac{15}{4 \cdot 15} + \frac{1}{5}$

Step 2.1.5

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

$\frac{20}{3 \cdot 20} + \frac{15}{4 \cdot 15} + \frac{1}{5} \cdot \frac{12}{12}$

Step 2.1.6

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

$\frac{20}{3 \cdot 20} + \frac{15}{4 \cdot 15} + \frac{12}{5 \cdot 12}$

Step 2.1.7

Multiply $3$ by $20$ .

$\frac{20}{60} + \frac{15}{4 \cdot 15} + \frac{12}{5 \cdot 12}$

Step 2.1.8

Multiply $4$ by $15$ .

$\frac{20}{60} + \frac{15}{60} + \frac{12}{5 \cdot 12}$

Step 2.1.9

Multiply $5$ by $12$ .

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

Step 2.2

Combine the numerators over the common denominator.

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

Step 2.3

Add $20$ and $15$ .

$\frac{35 + 12}{60}$

Step 2.4

Add $35$ and $12$ .

$\frac{47}{60}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\frac{47}{60}$

Decimal Form:

$0.78\overline{3}$