Calculus Examples

$\sum_{k = 1}^{4} {(- \frac{3}{4})}^{k}$

Step 1

Expand the series for each value of $k$ .

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

Step 2

Simplify.

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Step 2.1

Simplify each term.

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Step 2.1.1

Simplify.

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

Step 2.1.2

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 2.1.2.1

Apply the product rule to $- \frac{3}{4}$ .

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

Step 2.1.2.2

Apply the product rule to $\frac{3}{4}$ .

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

Step 2.1.3

Raise $- 1$ to the power of $2$ .

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

Step 2.1.4

Multiply $\frac{3^{2}}{4^{2}}$ by $1$ .

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

Step 2.1.5

Raise $3$ to the power of $2$ .

$- \frac{3}{4} + \frac{9}{4^{2}} + {(- \frac{3}{4})}^{3} + {(- \frac{3}{4})}^{4}$

Step 2.1.6

Raise $4$ to the power of $2$ .

$- \frac{3}{4} + \frac{9}{16} + {(- \frac{3}{4})}^{3} + {(- \frac{3}{4})}^{4}$

Step 2.1.7

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 2.1.7.1

Apply the product rule to $- \frac{3}{4}$ .

$- \frac{3}{4} + \frac{9}{16} + {(- 1)}^{3} {(\frac{3}{4})}^{3} + {(- \frac{3}{4})}^{4}$

Step 2.1.7.2

Apply the product rule to $\frac{3}{4}$ .

$- \frac{3}{4} + \frac{9}{16} + {(- 1)}^{3} \frac{3^{3}}{4^{3}} + {(- \frac{3}{4})}^{4}$

Step 2.1.8

Raise $- 1$ to the power of $3$ .

$- \frac{3}{4} + \frac{9}{16} - \frac{3^{3}}{4^{3}} + {(- \frac{3}{4})}^{4}$

Step 2.1.9

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

$- \frac{3}{4} + \frac{9}{16} - \frac{27}{4^{3}} + {(- \frac{3}{4})}^{4}$

Step 2.1.10

Raise $4$ to the power of $3$ .

$- \frac{3}{4} + \frac{9}{16} - \frac{27}{64} + {(- \frac{3}{4})}^{4}$

Step 2.1.11

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 2.1.11.1

Apply the product rule to $- \frac{3}{4}$ .

$- \frac{3}{4} + \frac{9}{16} - \frac{27}{64} + {(- 1)}^{4} {(\frac{3}{4})}^{4}$

Step 2.1.11.2

Apply the product rule to $\frac{3}{4}$ .

$- \frac{3}{4} + \frac{9}{16} - \frac{27}{64} + {(- 1)}^{4} \frac{3^{4}}{4^{4}}$

Step 2.1.12

Raise $- 1$ to the power of $4$ .

$- \frac{3}{4} + \frac{9}{16} - \frac{27}{64} + 1 \frac{3^{4}}{4^{4}}$

Step 2.1.13

Multiply $\frac{3^{4}}{4^{4}}$ by $1$ .

$- \frac{3}{4} + \frac{9}{16} - \frac{27}{64} + \frac{3^{4}}{4^{4}}$

Step 2.1.14

Raise $3$ to the power of $4$ .

$- \frac{3}{4} + \frac{9}{16} - \frac{27}{64} + \frac{81}{4^{4}}$

Step 2.1.15

Raise $4$ to the power of $4$ .

$- \frac{3}{4} + \frac{9}{16} - \frac{27}{64} + \frac{81}{256}$

Step 2.2

Find the common denominator.

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Step 2.2.1

Multiply $\frac{3}{4}$ by $\frac{64}{64}$ .

$- (\frac{3}{4} \cdot \frac{64}{64}) + \frac{9}{16} - \frac{27}{64} + \frac{81}{256}$

Step 2.2.2

Multiply $\frac{3}{4}$ by $\frac{64}{64}$ .

$- \frac{3 \cdot 64}{4 \cdot 64} + \frac{9}{16} - \frac{27}{64} + \frac{81}{256}$

Step 2.2.3

Multiply $\frac{9}{16}$ by $\frac{16}{16}$ .

$- \frac{3 \cdot 64}{4 \cdot 64} + \frac{9}{16} \cdot \frac{16}{16} - \frac{27}{64} + \frac{81}{256}$

Step 2.2.4

Multiply $\frac{9}{16}$ by $\frac{16}{16}$ .

$- \frac{3 \cdot 64}{4 \cdot 64} + \frac{9 \cdot 16}{16 \cdot 16} - \frac{27}{64} + \frac{81}{256}$

Step 2.2.5

Multiply $\frac{27}{64}$ by $\frac{4}{4}$ .

$- \frac{3 \cdot 64}{4 \cdot 64} + \frac{9 \cdot 16}{16 \cdot 16} - (\frac{27}{64} \cdot \frac{4}{4}) + \frac{81}{256}$

Step 2.2.6

Multiply $\frac{27}{64}$ by $\frac{4}{4}$ .

$- \frac{3 \cdot 64}{4 \cdot 64} + \frac{9 \cdot 16}{16 \cdot 16} - \frac{27 \cdot 4}{64 \cdot 4} + \frac{81}{256}$

Step 2.2.7

Multiply $4$ by $64$ .

$- \frac{3 \cdot 64}{256} + \frac{9 \cdot 16}{16 \cdot 16} - \frac{27 \cdot 4}{64 \cdot 4} + \frac{81}{256}$

Step 2.2.8

Multiply $16$ by $16$ .

$- \frac{3 \cdot 64}{256} + \frac{9 \cdot 16}{256} - \frac{27 \cdot 4}{64 \cdot 4} + \frac{81}{256}$

Step 2.2.9

Reorder the factors of $64 \cdot 4$ .

$- \frac{3 \cdot 64}{256} + \frac{9 \cdot 16}{256} - \frac{27 \cdot 4}{4 \cdot 64} + \frac{81}{256}$

Step 2.2.10

Multiply $4$ by $64$ .

$- \frac{3 \cdot 64}{256} + \frac{9 \cdot 16}{256} - \frac{27 \cdot 4}{256} + \frac{81}{256}$

Step 2.3

Combine the numerators over the common denominator.

$\frac{- 3 \cdot 64 + 9 \cdot 16 - 27 \cdot 4 + 81}{256}$

Step 2.4

Simplify each term.

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Step 2.4.1

Multiply $- 3$ by $64$ .

$\frac{- 192 + 9 \cdot 16 - 27 \cdot 4 + 81}{256}$

Step 2.4.2

Multiply $9$ by $16$ .

$\frac{- 192 + 144 - 27 \cdot 4 + 81}{256}$

Step 2.4.3

Multiply $- 27$ by $4$ .

$\frac{- 192 + 144 - 108 + 81}{256}$

Step 2.5

Add $- 192$ and $144$ .

$\frac{- 48 - 108 + 81}{256}$

Step 2.6

Subtract $108$ from $- 48$ .

$\frac{- 156 + 81}{256}$

Step 2.7

Add $- 156$ and $81$ .

$\frac{- 75}{256}$

Step 2.8

Move the negative in front of the fraction.

$- \frac{75}{256}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$- \frac{75}{256}$

Decimal Form:

$- 0.29296875$