Calculus Examples

$\sum_{1}^{4} \frac{7}{n}$

Step 1

Expand the series for each value of $n$ .

$\frac{7}{1} + \frac{7}{2} + \frac{7}{3} + \frac{7}{4}$

Step 2

Simplify.

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Step 2.1

Find the common denominator.

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Step 2.1.1

Multiply $\frac{7}{1}$ by $\frac{12}{12}$ .

$\frac{7}{1} \cdot \frac{12}{12} + \frac{7}{2} + \frac{7}{3} + \frac{7}{4}$

Step 2.1.2

Multiply $\frac{7}{1}$ by $\frac{12}{12}$ .

$\frac{7 \cdot 12}{12} + \frac{7}{2} + \frac{7}{3} + \frac{7}{4}$

Step 2.1.3

Multiply $\frac{7}{2}$ by $\frac{6}{6}$ .

$\frac{7 \cdot 12}{12} + \frac{7}{2} \cdot \frac{6}{6} + \frac{7}{3} + \frac{7}{4}$

Step 2.1.4

Multiply $\frac{7}{2}$ by $\frac{6}{6}$ .

$\frac{7 \cdot 12}{12} + \frac{7 \cdot 6}{2 \cdot 6} + \frac{7}{3} + \frac{7}{4}$

Step 2.1.5

Multiply $\frac{7}{3}$ by $\frac{4}{4}$ .

$\frac{7 \cdot 12}{12} + \frac{7 \cdot 6}{2 \cdot 6} + \frac{7}{3} \cdot \frac{4}{4} + \frac{7}{4}$

Step 2.1.6

Multiply $\frac{7}{3}$ by $\frac{4}{4}$ .

$\frac{7 \cdot 12}{12} + \frac{7 \cdot 6}{2 \cdot 6} + \frac{7 \cdot 4}{3 \cdot 4} + \frac{7}{4}$

Step 2.1.7

Multiply $\frac{7}{4}$ by $\frac{3}{3}$ .

$\frac{7 \cdot 12}{12} + \frac{7 \cdot 6}{2 \cdot 6} + \frac{7 \cdot 4}{3 \cdot 4} + \frac{7}{4} \cdot \frac{3}{3}$

Step 2.1.8

Multiply $\frac{7}{4}$ by $\frac{3}{3}$ .

$\frac{7 \cdot 12}{12} + \frac{7 \cdot 6}{2 \cdot 6} + \frac{7 \cdot 4}{3 \cdot 4} + \frac{7 \cdot 3}{4 \cdot 3}$

Step 2.1.9

Multiply $2$ by $6$ .

$\frac{7 \cdot 12}{12} + \frac{7 \cdot 6}{12} + \frac{7 \cdot 4}{3 \cdot 4} + \frac{7 \cdot 3}{4 \cdot 3}$

Step 2.1.10

Multiply $3$ by $4$ .

$\frac{7 \cdot 12}{12} + \frac{7 \cdot 6}{12} + \frac{7 \cdot 4}{12} + \frac{7 \cdot 3}{4 \cdot 3}$

Step 2.1.11

Reorder the factors of $4 \cdot 3$ .

$\frac{7 \cdot 12}{12} + \frac{7 \cdot 6}{12} + \frac{7 \cdot 4}{12} + \frac{7 \cdot 3}{3 \cdot 4}$

Step 2.1.12

Multiply $3$ by $4$ .

$\frac{7 \cdot 12}{12} + \frac{7 \cdot 6}{12} + \frac{7 \cdot 4}{12} + \frac{7 \cdot 3}{12}$

Step 2.2

Combine the numerators over the common denominator.

$\frac{7 \cdot 12 + 7 \cdot 6 + 7 \cdot 4 + 7 \cdot 3}{12}$

Step 2.3

Simplify each term.

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Step 2.3.1

Multiply $7$ by $12$ .

$\frac{84 + 7 \cdot 6 + 7 \cdot 4 + 7 \cdot 3}{12}$

Step 2.3.2

Multiply $7$ by $6$ .

$\frac{84 + 42 + 7 \cdot 4 + 7 \cdot 3}{12}$

Step 2.3.3

Multiply $7$ by $4$ .

$\frac{84 + 42 + 28 + 7 \cdot 3}{12}$

Step 2.3.4

Multiply $7$ by $3$ .

$\frac{84 + 42 + 28 + 21}{12}$

Step 2.4

Add $84$ and $42$ .

$\frac{126 + 28 + 21}{12}$

Step 2.5

Add $126$ and $28$ .

$\frac{154 + 21}{12}$

Step 2.6

Add $154$ and $21$ .

$\frac{175}{12}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\frac{175}{12}$

Decimal Form:

$14.58\overline{3}$

Mixed Number Form:

$14 \frac{7}{12}$