Calculus Examples

$\sum_{k = 3}^{4} \frac{1}{2 k}$

Step 1

Expand the series for each value of $k$ .

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

Step 2

Simplify.

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Step 2.1

Simplify each term.

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Step 2.1.1

Multiply $2$ by $3$ .

$\frac{1}{6} + \frac{1}{2 \cdot 4}$

Step 2.1.2

Multiply $2$ by $4$ .

$\frac{1}{6} + \frac{1}{8}$

Step 2.2

To write $\frac{1}{6}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{1}{6} \cdot \frac{4}{4} + \frac{1}{8}$

Step 2.3

To write $\frac{1}{8}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{1}{6} \cdot \frac{4}{4} + \frac{1}{8} \cdot \frac{3}{3}$

Step 2.4

Write each expression with a common denominator of $24$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.4.1

Multiply $\frac{1}{6}$ by $\frac{4}{4}$ .

$\frac{4}{6 \cdot 4} + \frac{1}{8} \cdot \frac{3}{3}$

Step 2.4.2

Multiply $6$ by $4$ .

$\frac{4}{24} + \frac{1}{8} \cdot \frac{3}{3}$

Step 2.4.3

Multiply $\frac{1}{8}$ by $\frac{3}{3}$ .

$\frac{4}{24} + \frac{3}{8 \cdot 3}$

Step 2.4.4

Multiply $8$ by $3$ .

$\frac{4}{24} + \frac{3}{24}$

Step 2.5

Combine the numerators over the common denominator.

$\frac{4 + 3}{24}$

Step 2.6

Add $4$ and $3$ .

$\frac{7}{24}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\frac{7}{24}$

Decimal Form:

$0.291\overline{6}$