Calculus Examples

$\sum_{i = 1}^{\infty} \frac{2}{3^{i}}$

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 $r = \frac{a_{i + 1}}{a_{i}}$ and simplifying.

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Step 2.1

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

$r = \frac{\frac{2}{3^{i + 1}}}{\frac{2}{3^{i}}}$

Step 2.2

Simplify.

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Step 2.2.1

Multiply the numerator by the reciprocal of the denominator.

$r = \frac{2}{3^{i + 1}} \cdot \frac{3^{i}}{2}$

Step 2.2.2

Combine.

$r = \frac{2 \cdot 3^{i}}{3^{i + 1} \cdot 2}$

Step 2.2.3

Cancel the common factor of $2$ .

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Step 2.2.3.1

Cancel the common factor.

$r = \frac{2 \cdot 3^{i}}{3^{i + 1} \cdot 2}$

Step 2.2.3.2

Rewrite the expression.

$r = \frac{3^{i}}{3^{i + 1}}$

Step 2.2.4

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

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Step 2.2.4.1

Factor $3^{i + 1}$ out of $3^{i}$ .

$r = \frac{3^{i + 1} \cdot 3^{i - (i + 1)}}{3^{i + 1}}$

Step 2.2.4.2

Cancel the common factors.

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Step 2.2.4.2.1

Multiply by $1$ .

$r = \frac{3^{i + 1} \cdot 3^{i - (i + 1)}}{3^{i + 1} \cdot 1}$

Step 2.2.4.2.2

Cancel the common factor.

$r = \frac{3^{i + 1} \cdot 3^{i - (i + 1)}}{3^{i + 1} \cdot 1}$

Step 2.2.4.2.3

Rewrite the expression.

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

Step 2.2.4.2.4

Divide $3^{i - (i + 1)}$ by $1$ .

$r = 3^{i - (i + 1)}$

Step 2.2.5

Simplify each term.

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Step 2.2.5.1

Apply the distributive property.

$r = 3^{i - i - 1 \cdot 1}$

Step 2.2.5.2

Multiply $- 1$ by $1$ .

$r = 3^{i - i - 1}$

Step 2.2.6

Subtract $i$ from $i$ .

$r = 3^{0 - 1}$

Step 2.2.7

Subtract $1$ from $0$ .

$r = 3^{- 1}$

Step 2.2.8

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

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

Step 3

Since $| r | < 1$ , the series converges.

Step 4

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

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Step 4.1

Substitute $1$ for $i$ into $\frac{2}{3^{i}}$ .

$a = \frac{2}{3^{1}}$

Step 4.2

Evaluate the exponent.

$a = \frac{2}{3}$

Step 5

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

$\frac{\frac{2}{3}}{1 - \frac{1}{3}}$

Step 6

Simplify.

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Step 6.1

Multiply the numerator by the reciprocal of the denominator.

$\frac{2}{3} \cdot \frac{1}{1 - \frac{1}{3}}$

Step 6.2

Simplify the denominator.

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Step 6.2.1

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

$\frac{2}{3} \cdot \frac{1}{\frac{3}{3} - \frac{1}{3}}$

Step 6.2.2

Combine the numerators over the common denominator.

$\frac{2}{3} \cdot \frac{1}{\frac{3 - 1}{3}}$

Step 6.2.3

Subtract $1$ from $3$ .

$\frac{2}{3} \cdot \frac{1}{\frac{2}{3}}$

Step 6.3

Multiply the numerator by the reciprocal of the denominator.

$\frac{2}{3} (1 (\frac{3}{2}))$

Step 6.4

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

$\frac{2}{3} \cdot \frac{3}{2}$

Step 6.5

Cancel the common factor of $2$ .

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Step 6.5.1

Cancel the common factor.

$\frac{2}{3} \cdot \frac{3}{2}$

Step 6.5.2

Rewrite the expression.

$\frac{1}{3} \cdot 3$

Step 6.6

Cancel the common factor of $3$ .

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Step 6.6.1

Cancel the common factor.

$\frac{1}{3} \cdot 3$

Step 6.6.2

Rewrite the expression.

$1$