Calculus Examples

$\sum_{n = 0}^{\infty} \frac{1}{6^{n}}$

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

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

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

$r = \frac{\frac{1}{6^{n + 1}}}{\frac{1}{6^{n}}}$

Simplify.

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Multiply the numerator by the reciprocal of the denominator.

$r = \frac{1}{6^{n + 1}} \cdot 6^{n}$

Combine $\frac{1}{6^{n + 1}}$ and $6^{n}$ .

$r = \frac{6^{n}}{6^{n + 1}}$

Cancel the common factor of $6^{n}$ and $6^{n + 1}$ .

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Factor $6^{n + 1}$ out of $6^{n}$ .

$r = \frac{6^{n + 1} \cdot 6^{n - (n + 1)}}{6^{n + 1}}$

Cancel the common factors.

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Multiply by $1$ .

$r = \frac{6^{n + 1} \cdot 6^{n - (n + 1)}}{6^{n + 1} \cdot 1}$

Cancel the common factor.

$r = \frac{6^{n + 1} \cdot 6^{n - (n + 1)}}{6^{n + 1} \cdot 1}$

Rewrite the expression.

$r = \frac{6^{n - (n + 1)}}{1}$

Divide $6^{n - (n + 1)}$ by $1$ .

$r = 6^{n - (n + 1)}$

Simplify each term.

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Apply the distributive property.

$r = 6^{n - n - 1 \cdot 1}$

Multiply $- 1$ by $1$ .

$r = 6^{n - n - 1}$

Subtract $n$ from $n$ .

$r = 6^{0 - 1}$

Subtract $1$ from $0$ .

$r = 6^{- 1}$

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$r = \frac{1}{6}$

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

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Substitute $0$ for $n$ into $\frac{1}{6^{n}}$ .

$a = \frac{1}{6^{0}}$

Simplify.

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Anything raised to $0$ is $1$ .

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

Divide $1$ by $1$ .

$a = 1$

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

$\frac{1}{1 - \frac{1}{6}}$

Simplify.

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Simplify the denominator.

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

$\frac{1}{\frac{6}{6} - \frac{1}{6}}$

Combine the numerators over the common denominator.

$\frac{1}{\frac{6 - 1}{6}}$

Subtract $1$ from $6$ .

$\frac{1}{\frac{5}{6}}$

Multiply the numerator by the reciprocal of the denominator.

$1 (\frac{6}{5})$

Multiply $\frac{6}{5}$ by $1$ .

$\frac{6}{5}$

The result can be shown in multiple forms.

Exact Form:

$\frac{6}{5}$

Decimal Form:

$1.2$

Mixed Number Form:

$1 \frac{1}{5}$