Calculus Examples

$\sum_{n = 1}^{\infty} \frac{7^{n}}{4^{n + 3}}$

Step 1

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.

Step 2

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

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Step 2.1

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

$r = \frac{\frac{7^{n + 1}}{4^{n + 1 + 3}}}{\frac{7^{n}}{4^{n + 3}}}$

Step 2.2

Simplify.

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Step 2.2.1

Multiply the numerator by the reciprocal of the denominator.

$r = \frac{7^{n + 1}}{4^{n + 1 + 3}} \cdot \frac{4^{n + 3}}{7^{n}}$

Step 2.2.2

Combine.

$r = \frac{7^{n + 1} \cdot 4^{n + 3}}{4^{n + 1 + 3} \cdot 7^{n}}$

Step 2.2.3

Cancel the common factor of $7^{n + 1}$ and $7^{n}$ .

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Step 2.2.3.1

Factor $7^{n}$ out of $7^{n + 1} \cdot 4^{n + 3}$ .

$r = \frac{7^{n} (7 \cdot 4^{n + 3})}{4^{n + 1 + 3} \cdot 7^{n}}$

Step 2.2.3.2

Cancel the common factors.

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Step 2.2.3.2.1

Factor $7^{n}$ out of $4^{n + 1 + 3} \cdot 7^{n}$ .

$r = \frac{7^{n} (7 \cdot 4^{n + 3})}{7^{n} \cdot 4^{n + 1 + 3}}$

Step 2.2.3.2.2

Cancel the common factor.

$r = \frac{7^{n} (7 \cdot 4^{n + 3})}{7^{n} \cdot 4^{n + 1 + 3}}$

Step 2.2.3.2.3

Rewrite the expression.

$r = \frac{7 \cdot 4^{n + 3}}{4^{n + 1 + 3}}$

Step 2.2.4

Cancel the common factor of $4^{n + 3}$ and $4^{n + 1 + 3}$ .

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Step 2.2.4.1

Factor $4^{n + 1 + 3}$ out of $7 \cdot 4^{n + 3}$ .

$r = \frac{4^{n + 1 + 3} (7 \cdot 4^{n + 3 - (n + 1 + 3)})}{4^{n + 1 + 3}}$

Step 2.2.4.2

Cancel the common factors.

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Step 2.2.4.2.1

Multiply by $1$ .

$r = \frac{4^{n + 1 + 3} (7 \cdot 4^{n + 3 - (n + 1 + 3)})}{4^{n + 1 + 3} \cdot 1}$

Step 2.2.4.2.2

Cancel the common factor.

$r = \frac{4^{n + 1 + 3} (7 \cdot 4^{n + 3 - (n + 1 + 3)})}{4^{n + 1 + 3} \cdot 1}$

Step 2.2.4.2.3

Rewrite the expression.

$r = \frac{7 \cdot 4^{n + 3 - (n + 1 + 3)}}{1}$

Step 2.2.4.2.4

Divide $7 \cdot 4^{n + 3 - (n + 1 + 3)}$ by $1$ .

$r = 7 \cdot 4^{n + 3 - (n + 1 + 3)}$

Step 2.2.5

Simplify each term.

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Step 2.2.5.1

Add $1$ and $3$ .

$r = 7 \cdot 4^{n + 3 - (n + 4)}$

Step 2.2.5.2

Apply the distributive property.

$r = 7 \cdot 4^{n + 3 - n - 1 \cdot 4}$

Step 2.2.5.3

Multiply $- 1$ by $4$ .

$r = 7 \cdot 4^{n + 3 - n - 4}$

Step 2.2.6

Combine the opposite terms in $n + 3 - n - 4$ .

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Step 2.2.6.1

Subtract $n$ from $n$ .

$r = 7 \cdot 4^{0 + 3 - 4}$

Step 2.2.6.2

Add $0$ and $3$ .

$r = 7 \cdot 4^{3 - 4}$

Step 2.2.7

Subtract $4$ from $3$ .

$r = 7 \cdot 4^{- 1}$

Step 2.2.8

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

$r = 7 \cdot \frac{1}{4}$

Step 2.2.9

Combine $7$ and $\frac{1}{4}$ .

$r = \frac{7}{4}$

Step 3

Check if the series is convergent or divergent.

Since $| r | \geq 1$ , the series diverges.