Calculus Examples

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

Step 1

Expand the series for each value of $k$ .

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

Step 2

Simplify.

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Step 2.1

Find the common denominator.

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Step 2.1.1

Multiply $\frac{1}{1}$ by $\frac{6}{6}$ .

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

Step 2.1.2

Multiply $\frac{1}{1}$ by $\frac{6}{6}$ .

$\frac{6}{6} + \frac{1}{2} + \frac{1}{3}$

Step 2.1.3

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

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

Step 2.1.4

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

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

Step 2.1.5

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

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

Step 2.1.6

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

$\frac{6}{6} + \frac{3}{2 \cdot 3} + \frac{2}{3 \cdot 2}$

Step 2.1.7

Multiply $2$ by $3$ .

$\frac{6}{6} + \frac{3}{6} + \frac{2}{3 \cdot 2}$

Step 2.1.8

Reorder the factors of $3 \cdot 2$ .

$\frac{6}{6} + \frac{3}{6} + \frac{2}{2 \cdot 3}$

Step 2.1.9

Multiply $2$ by $3$ .

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

Step 2.2

Combine the numerators over the common denominator.

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

Step 2.3

Add $6$ and $3$ .

$\frac{9 + 2}{6}$

Step 2.4

Add $9$ and $2$ .

$\frac{11}{6}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\frac{11}{6}$

Decimal Form:

$1.8\overline{3}$

Mixed Number Form:

$1 \frac{5}{6}$