Calculus Examples

$\sum_{n = 1}^{4} {(\frac{1}{6})}^{n - 1}$

Step 1

Expand the series for each value of $n$ .

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

Step 2

Simplify.

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Step 2.1

Simplify each term.

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Step 2.1.1

Subtract $1$ from $1$ .

${(\frac{1}{6})}^{0} + {(\frac{1}{6})}^{2 - 1} + {(\frac{1}{6})}^{3 - 1} + {(\frac{1}{6})}^{4 - 1}$

Step 2.1.2

Apply the product rule to $\frac{1}{6}$ .

$\frac{1^{0}}{6^{0}} + {(\frac{1}{6})}^{2 - 1} + {(\frac{1}{6})}^{3 - 1} + {(\frac{1}{6})}^{4 - 1}$

Step 2.1.3

Anything raised to $0$ is $1$ .

$\frac{1}{6^{0}} + {(\frac{1}{6})}^{2 - 1} + {(\frac{1}{6})}^{3 - 1} + {(\frac{1}{6})}^{4 - 1}$

Step 2.1.4

Anything raised to $0$ is $1$ .

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

Step 2.1.5

Divide $1$ by $1$ .

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

Step 2.1.6

Subtract $1$ from $2$ .

$1 + {(\frac{1}{6})}^{1} + {(\frac{1}{6})}^{3 - 1} + {(\frac{1}{6})}^{4 - 1}$

Step 2.1.7

Simplify.

$1 + \frac{1}{6} + {(\frac{1}{6})}^{3 - 1} + {(\frac{1}{6})}^{4 - 1}$

Step 2.1.8

Subtract $1$ from $3$ .

$1 + \frac{1}{6} + {(\frac{1}{6})}^{2} + {(\frac{1}{6})}^{4 - 1}$

Step 2.1.9

Apply the product rule to $\frac{1}{6}$ .

$1 + \frac{1}{6} + \frac{1^{2}}{6^{2}} + {(\frac{1}{6})}^{4 - 1}$

Step 2.1.10

One to any power is one.

$1 + \frac{1}{6} + \frac{1}{6^{2}} + {(\frac{1}{6})}^{4 - 1}$

Step 2.1.11

Raise $6$ to the power of $2$ .

$1 + \frac{1}{6} + \frac{1}{36} + {(\frac{1}{6})}^{4 - 1}$

Step 2.1.12

Subtract $1$ from $4$ .

$1 + \frac{1}{6} + \frac{1}{36} + {(\frac{1}{6})}^{3}$

Step 2.1.13

Apply the product rule to $\frac{1}{6}$ .

$1 + \frac{1}{6} + \frac{1}{36} + \frac{1^{3}}{6^{3}}$

Step 2.1.14

One to any power is one.

$1 + \frac{1}{6} + \frac{1}{36} + \frac{1}{6^{3}}$

Step 2.1.15

Raise $6$ to the power of $3$ .

$1 + \frac{1}{6} + \frac{1}{36} + \frac{1}{216}$

Step 2.2

Find the common denominator.

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Step 2.2.1

Write $1$ as a fraction with denominator $1$ .

$\frac{1}{1} + \frac{1}{6} + \frac{1}{36} + \frac{1}{216}$

Step 2.2.2

Multiply $\frac{1}{1}$ by $\frac{216}{216}$ .

$\frac{1}{1} \cdot \frac{216}{216} + \frac{1}{6} + \frac{1}{36} + \frac{1}{216}$

Step 2.2.3

Multiply $\frac{1}{1}$ by $\frac{216}{216}$ .

$\frac{216}{216} + \frac{1}{6} + \frac{1}{36} + \frac{1}{216}$

Step 2.2.4

Multiply $\frac{1}{6}$ by $\frac{36}{36}$ .

$\frac{216}{216} + \frac{1}{6} \cdot \frac{36}{36} + \frac{1}{36} + \frac{1}{216}$

Step 2.2.5

Multiply $\frac{1}{6}$ by $\frac{36}{36}$ .

$\frac{216}{216} + \frac{36}{6 \cdot 36} + \frac{1}{36} + \frac{1}{216}$

Step 2.2.6

Multiply $\frac{1}{36}$ by $\frac{6}{6}$ .

$\frac{216}{216} + \frac{36}{6 \cdot 36} + \frac{1}{36} \cdot \frac{6}{6} + \frac{1}{216}$

Step 2.2.7

Multiply $\frac{1}{36}$ by $\frac{6}{6}$ .

$\frac{216}{216} + \frac{36}{6 \cdot 36} + \frac{6}{36 \cdot 6} + \frac{1}{216}$

Step 2.2.8

Multiply $6$ by $36$ .

$\frac{216}{216} + \frac{36}{216} + \frac{6}{36 \cdot 6} + \frac{1}{216}$

Step 2.2.9

Reorder the factors of $36 \cdot 6$ .

$\frac{216}{216} + \frac{36}{216} + \frac{6}{6 \cdot 36} + \frac{1}{216}$

Step 2.2.10

Multiply $6$ by $36$ .

$\frac{216}{216} + \frac{36}{216} + \frac{6}{216} + \frac{1}{216}$

Step 2.3

Combine the numerators over the common denominator.

$\frac{216 + 36 + 6 + 1}{216}$

Step 2.4

Add $216$ and $36$ .

$\frac{252 + 6 + 1}{216}$

Step 2.5

Add $252$ and $6$ .

$\frac{258 + 1}{216}$

Step 2.6

Add $258$ and $1$ .

$\frac{259}{216}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\frac{259}{216}$

Decimal Form:

$1.199\overline{074}$

Mixed Number Form:

$1 \frac{43}{216}$