Calculus Examples

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

Step 1

Expand the series for each value of $n$ .

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

Step 2

Simplify.

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Step 2.1

Add $1$ and $1$ .

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

Step 2.2

Add $2$ and $1$ .

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

Step 2.3

Add $3$ and $1$ .

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

Step 2.4

Add $4$ and $1$ .

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

Step 2.5

Simplify each term.

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Step 2.5.1

Add $1$ and $4$ .

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

Step 2.5.2

Add $2$ and $4$ .

$\frac{2}{5} + \frac{3}{6} + \frac{4}{3 + 4} + \frac{5}{4 + 4}$

Step 2.5.3

Cancel the common factor of $3$ and $6$ .

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Step 2.5.3.1

Factor $3$ out of $3$ .

$\frac{2}{5} + \frac{3 (1)}{6} + \frac{4}{3 + 4} + \frac{5}{4 + 4}$

Step 2.5.3.2

Cancel the common factors.

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Step 2.5.3.2.1

Factor $3$ out of $6$ .

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

Step 2.5.3.2.2

Cancel the common factor.

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

Step 2.5.3.2.3

Rewrite the expression.

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

Step 2.5.4

Add $3$ and $4$ .

$\frac{2}{5} + \frac{1}{2} + \frac{4}{7} + \frac{5}{4 + 4}$

Step 2.5.5

Add $4$ and $4$ .

$\frac{2}{5} + \frac{1}{2} + \frac{4}{7} + \frac{5}{8}$

Step 2.6

Find the common denominator.

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Step 2.6.1

Multiply $\frac{2}{5}$ by $\frac{56}{56}$ .

$\frac{2}{5} \cdot \frac{56}{56} + \frac{1}{2} + \frac{4}{7} + \frac{5}{8}$

Step 2.6.2

Multiply $\frac{2}{5}$ by $\frac{56}{56}$ .

$\frac{2 \cdot 56}{5 \cdot 56} + \frac{1}{2} + \frac{4}{7} + \frac{5}{8}$

Step 2.6.3

Multiply $\frac{1}{2}$ by $\frac{140}{140}$ .

$\frac{2 \cdot 56}{5 \cdot 56} + \frac{1}{2} \cdot \frac{140}{140} + \frac{4}{7} + \frac{5}{8}$

Step 2.6.4

Multiply $\frac{1}{2}$ by $\frac{140}{140}$ .

$\frac{2 \cdot 56}{5 \cdot 56} + \frac{140}{2 \cdot 140} + \frac{4}{7} + \frac{5}{8}$

Step 2.6.5

Multiply $\frac{4}{7}$ by $\frac{40}{40}$ .

$\frac{2 \cdot 56}{5 \cdot 56} + \frac{140}{2 \cdot 140} + \frac{4}{7} \cdot \frac{40}{40} + \frac{5}{8}$

Step 2.6.6

Multiply $\frac{4}{7}$ by $\frac{40}{40}$ .

$\frac{2 \cdot 56}{5 \cdot 56} + \frac{140}{2 \cdot 140} + \frac{4 \cdot 40}{7 \cdot 40} + \frac{5}{8}$

Step 2.6.7

Multiply $\frac{5}{8}$ by $\frac{35}{35}$ .

$\frac{2 \cdot 56}{5 \cdot 56} + \frac{140}{2 \cdot 140} + \frac{4 \cdot 40}{7 \cdot 40} + \frac{5}{8} \cdot \frac{35}{35}$

Step 2.6.8

Multiply $\frac{5}{8}$ by $\frac{35}{35}$ .

$\frac{2 \cdot 56}{5 \cdot 56} + \frac{140}{2 \cdot 140} + \frac{4 \cdot 40}{7 \cdot 40} + \frac{5 \cdot 35}{8 \cdot 35}$

Step 2.6.9

Multiply $5$ by $56$ .

$\frac{2 \cdot 56}{280} + \frac{140}{2 \cdot 140} + \frac{4 \cdot 40}{7 \cdot 40} + \frac{5 \cdot 35}{8 \cdot 35}$

Step 2.6.10

Multiply $2$ by $140$ .

$\frac{2 \cdot 56}{280} + \frac{140}{280} + \frac{4 \cdot 40}{7 \cdot 40} + \frac{5 \cdot 35}{8 \cdot 35}$

Step 2.6.11

Multiply $7$ by $40$ .

$\frac{2 \cdot 56}{280} + \frac{140}{280} + \frac{4 \cdot 40}{280} + \frac{5 \cdot 35}{8 \cdot 35}$

Step 2.6.12

Multiply $8$ by $35$ .

$\frac{2 \cdot 56}{280} + \frac{140}{280} + \frac{4 \cdot 40}{280} + \frac{5 \cdot 35}{280}$

Step 2.7

Combine the numerators over the common denominator.

$\frac{2 \cdot 56 + 140 + 4 \cdot 40 + 5 \cdot 35}{280}$

Step 2.8

Simplify each term.

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Step 2.8.1

Multiply $2$ by $56$ .

$\frac{112 + 140 + 4 \cdot 40 + 5 \cdot 35}{280}$

Step 2.8.2

Multiply $4$ by $40$ .

$\frac{112 + 140 + 160 + 5 \cdot 35}{280}$

Step 2.8.3

Multiply $5$ by $35$ .

$\frac{112 + 140 + 160 + 175}{280}$

Step 2.9

Add $112$ and $140$ .

$\frac{252 + 160 + 175}{280}$

Step 2.10

Add $252$ and $160$ .

$\frac{412 + 175}{280}$

Step 2.11

Add $412$ and $175$ .

$\frac{587}{280}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\frac{587}{280}$

Decimal Form:

$2.09642857 \dots$

Mixed Number Form:

$2 \frac{27}{280}$