Precalculus Examples

$\sum_{i = 1}^{4} \frac{1}{i + 4}$

Step 1

Expand the series for each value of $i$ .

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

Step 2

Simplify.

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Step 2.1

Simplify each term.

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Step 2.1.1

Add $1$ and $4$ .

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

Step 2.1.2

Add $2$ and $4$ .

$\frac{1}{5} + \frac{1}{6} + \frac{1}{3 + 4} + \frac{1}{4 + 4}$

Step 2.1.3

Add $3$ and $4$ .

$\frac{1}{5} + \frac{1}{6} + \frac{1}{7} + \frac{1}{4 + 4}$

Step 2.1.4

Add $4$ and $4$ .

$\frac{1}{5} + \frac{1}{6} + \frac{1}{7} + \frac{1}{8}$

Step 2.2

Find the common denominator.

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Step 2.2.1

Multiply $\frac{1}{5}$ by $\frac{168}{168}$ .

$\frac{1}{5} \cdot \frac{168}{168} + \frac{1}{6} + \frac{1}{7} + \frac{1}{8}$

Step 2.2.2

Multiply $\frac{1}{5}$ by $\frac{168}{168}$ .

$\frac{168}{5 \cdot 168} + \frac{1}{6} + \frac{1}{7} + \frac{1}{8}$

Step 2.2.3

Multiply $\frac{1}{6}$ by $\frac{140}{140}$ .

$\frac{168}{5 \cdot 168} + \frac{1}{6} \cdot \frac{140}{140} + \frac{1}{7} + \frac{1}{8}$

Step 2.2.4

Multiply $\frac{1}{6}$ by $\frac{140}{140}$ .

$\frac{168}{5 \cdot 168} + \frac{140}{6 \cdot 140} + \frac{1}{7} + \frac{1}{8}$

Step 2.2.5

Multiply $\frac{1}{7}$ by $\frac{120}{120}$ .

$\frac{168}{5 \cdot 168} + \frac{140}{6 \cdot 140} + \frac{1}{7} \cdot \frac{120}{120} + \frac{1}{8}$

Step 2.2.6

Multiply $\frac{1}{7}$ by $\frac{120}{120}$ .

$\frac{168}{5 \cdot 168} + \frac{140}{6 \cdot 140} + \frac{120}{7 \cdot 120} + \frac{1}{8}$

Step 2.2.7

Multiply $\frac{1}{8}$ by $\frac{105}{105}$ .

$\frac{168}{5 \cdot 168} + \frac{140}{6 \cdot 140} + \frac{120}{7 \cdot 120} + \frac{1}{8} \cdot \frac{105}{105}$

Step 2.2.8

Multiply $\frac{1}{8}$ by $\frac{105}{105}$ .

$\frac{168}{5 \cdot 168} + \frac{140}{6 \cdot 140} + \frac{120}{7 \cdot 120} + \frac{105}{8 \cdot 105}$

Step 2.2.9

Multiply $5$ by $168$ .

$\frac{168}{840} + \frac{140}{6 \cdot 140} + \frac{120}{7 \cdot 120} + \frac{105}{8 \cdot 105}$

Step 2.2.10

Multiply $6$ by $140$ .

$\frac{168}{840} + \frac{140}{840} + \frac{120}{7 \cdot 120} + \frac{105}{8 \cdot 105}$

Step 2.2.11

Multiply $7$ by $120$ .

$\frac{168}{840} + \frac{140}{840} + \frac{120}{840} + \frac{105}{8 \cdot 105}$

Step 2.2.12

Multiply $8$ by $105$ .

$\frac{168}{840} + \frac{140}{840} + \frac{120}{840} + \frac{105}{840}$

Step 2.3

Combine the numerators over the common denominator.

$\frac{168 + 140 + 120 + 105}{840}$

Step 2.4

Add $168$ and $140$ .

$\frac{308 + 120 + 105}{840}$

Step 2.5

Add $308$ and $120$ .

$\frac{428 + 105}{840}$

Step 2.6

Add $428$ and $105$ .

$\frac{533}{840}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\frac{533}{840}$

Decimal Form:

$0.63452380 \dots$