Calculus Examples

$\sum_{k = 0}^{9} {(- \frac{3}{5})}^{k}$

Step 1

The sum of a finite geometric series can be found using the formula $a (\frac{1 - r^{n}}{1 - r})$ where $a$ is the first term and $r$ is the ratio between successive terms.

Step 2

Find the ratio of successive terms by plugging into the formula $r = \frac{a_{k + 1}}{a_{k}}$ and simplifying.

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Step 2.1

Substitute $a_{k}$ and $a_{k + 1}$ into the formula for $r$ .

$r = \frac{{(- \frac{3}{5})}^{k + 1}}{{(- \frac{3}{5})}^{k}}$

Step 2.2

Cancel the common factor of ${(- \frac{3}{5})}^{k + 1}$ and ${(- \frac{3}{5})}^{k}$ .

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Step 2.2.1

Factor ${(- \frac{3}{5})}^{k}$ out of ${(- \frac{3}{5})}^{k + 1}$ .

$r = \frac{{(- \frac{3}{5})}^{k} (- \frac{3}{5})}{{(- \frac{3}{5})}^{k}}$

Step 2.2.2

Cancel the common factors.

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Step 2.2.2.1

Multiply by $1$ .

$r = \frac{{(- \frac{3}{5})}^{k} (- \frac{3}{5})}{{(- \frac{3}{5})}^{k} \cdot 1}$

Step 2.2.2.2

Cancel the common factor.

$r = \frac{{(- \frac{3}{5})}^{k} (- \frac{3}{5})}{{(- \frac{3}{5})}^{k} \cdot 1}$

Step 2.2.2.3

Rewrite the expression.

$r = \frac{- \frac{3}{5}}{1}$

Step 2.2.2.4

Divide $- \frac{3}{5}$ by $1$ .

$r = - \frac{3}{5}$

Step 3

Find the first term in the series by substituting in the lower bound and simplifying.

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Step 3.1

Substitute $0$ for $k$ into ${(- \frac{3}{5})}^{k}$ .

$a = {(- \frac{3}{5})}^{0}$

Step 3.2

Simplify.

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Step 3.2.1

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

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Step 3.2.1.1

Apply the product rule to $- \frac{3}{5}$ .

$a = {(- 1)}^{0} {(\frac{3}{5})}^{0}$

Step 3.2.1.2

Apply the product rule to $\frac{3}{5}$ .

$a = {(- 1)}^{0} \frac{3^{0}}{5^{0}}$

Step 3.2.2

Anything raised to $0$ is $1$ .

$a = 1 \frac{3^{0}}{5^{0}}$

Step 3.2.3

Multiply $\frac{3^{0}}{5^{0}}$ by $1$ .

$a = \frac{3^{0}}{5^{0}}$

Step 3.2.4

Anything raised to $0$ is $1$ .

$a = \frac{1}{5^{0}}$

Step 3.2.5

Anything raised to $0$ is $1$ .

$a = \frac{1}{1}$

Step 3.2.6

Divide $1$ by $1$ .

$a = 1$

Step 4

Substitute the values of the ratio, first term, and number of terms into the sum formula.

$1 \frac{1 - {(- \frac{3}{5})}^{10}}{1 - - \frac{3}{5}}$

Step 5

Simplify.

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Step 5.1

Multiply $\frac{1 - {(- \frac{3}{5})}^{10}}{1 - - \frac{3}{5}}$ by $1$ .

$\frac{1 - {(- \frac{3}{5})}^{10}}{1 - - \frac{3}{5}}$

Step 5.2

Simplify the numerator.

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Step 5.2.1

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

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Step 5.2.1.1

Apply the product rule to $- \frac{3}{5}$ .

$\frac{1 - ({(- 1)}^{10} {(\frac{3}{5})}^{10})}{1 - - \frac{3}{5}}$

Step 5.2.1.2

Apply the product rule to $\frac{3}{5}$ .

$\frac{1 - ({(- 1)}^{10} \frac{3^{10}}{5^{10}})}{1 - - \frac{3}{5}}$

Step 5.2.2

Multiply $- 1$ by ${(- 1)}^{10}$ by adding the exponents.

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Step 5.2.2.1

Move ${(- 1)}^{10}$ .

$\frac{1 + {(- 1)}^{10} \cdot - 1 \frac{3^{10}}{5^{10}}}{1 - - \frac{3}{5}}$

Step 5.2.2.2

Multiply ${(- 1)}^{10}$ by $- 1$ .

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Step 5.2.2.2.1

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

$\frac{1 + {(- 1)}^{10} \cdot {(- 1)}^{1} \frac{3^{10}}{5^{10}}}{1 - - \frac{3}{5}}$

Step 5.2.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{1 + {(- 1)}^{10 + 1} \frac{3^{10}}{5^{10}}}{1 - - \frac{3}{5}}$

Step 5.2.2.3

Add $10$ and $1$ .

$\frac{1 + {(- 1)}^{11} \frac{3^{10}}{5^{10}}}{1 - - \frac{3}{5}}$

Step 5.2.3

Raise $- 1$ to the power of $11$ .

$\frac{1 - \frac{3^{10}}{5^{10}}}{1 - - \frac{3}{5}}$

Step 5.2.4

Raise $3$ to the power of $10$ .

$\frac{1 - \frac{59049}{5^{10}}}{1 - - \frac{3}{5}}$

Step 5.2.5

Raise $5$ to the power of $10$ .

$\frac{1 - \frac{59049}{9765625}}{1 - - \frac{3}{5}}$

Step 5.2.6

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

$\frac{\frac{9765625}{9765625} - \frac{59049}{9765625}}{1 - - \frac{3}{5}}$

Step 5.2.7

Combine the numerators over the common denominator.

$\frac{\frac{9765625 - 59049}{9765625}}{1 - - \frac{3}{5}}$

Step 5.2.8

Subtract $59049$ from $9765625$ .

$\frac{\frac{9706576}{9765625}}{1 - - \frac{3}{5}}$

Step 5.3

Simplify the denominator.

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Step 5.3.1

Multiply $- - \frac{3}{5}$ .

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Step 5.3.1.1

Multiply $- 1$ by $- 1$ .

$\frac{\frac{9706576}{9765625}}{1 + 1 (\frac{3}{5})}$

Step 5.3.1.2

Multiply $\frac{3}{5}$ by $1$ .

$\frac{\frac{9706576}{9765625}}{1 + \frac{3}{5}}$

Step 5.3.2

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

$\frac{\frac{9706576}{9765625}}{\frac{5}{5} + \frac{3}{5}}$

Step 5.3.3

Combine the numerators over the common denominator.

$\frac{\frac{9706576}{9765625}}{\frac{5 + 3}{5}}$

Step 5.3.4

Add $5$ and $3$ .

$\frac{\frac{9706576}{9765625}}{\frac{8}{5}}$

Step 5.4

Multiply the numerator by the reciprocal of the denominator.

$\frac{9706576}{9765625} \cdot \frac{5}{8}$

Step 5.5

Cancel the common factor of $8$ .

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Step 5.5.1

Factor $8$ out of $9706576$ .

$\frac{8 (1213322)}{9765625} \cdot \frac{5}{8}$

Step 5.5.2

Cancel the common factor.

$\frac{8 \cdot 1213322}{9765625} \cdot \frac{5}{8}$

Step 5.5.3

Rewrite the expression.

$\frac{1213322}{9765625} \cdot 5$

Step 5.6

Cancel the common factor of $5$ .

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Step 5.6.1

Factor $5$ out of $9765625$ .

$\frac{1213322}{5 (1953125)} \cdot 5$

Step 5.6.2

Cancel the common factor.

$\frac{1213322}{5 \cdot 1953125} \cdot 5$

Step 5.6.3

Rewrite the expression.

$\frac{1213322}{1953125}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{1213322}{1953125}$

Decimal Form:

$0.62122086 \dots$