Calculus Examples

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

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 $r$ .

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

Step 2.2

Simplify.

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Step 2.2.1

Cancel the common factor of $\frac{1}{2}$ .

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Step 2.2.1.1

Cancel the common factor.

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

Step 2.2.1.2

Rewrite the expression.

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

Step 2.2.2

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

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Step 2.2.2.1

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

$r = \frac{{(\frac{1}{3})}^{i - 1} {(\frac{1}{3})}^{i + 0 - (i - 1)}}{{(\frac{1}{3})}^{i - 1}}$

Step 2.2.2.2

Cancel the common factors.

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Step 2.2.2.2.1

Multiply by $1$ .

$r = \frac{{(\frac{1}{3})}^{i - 1} {(\frac{1}{3})}^{i + 0 - (i - 1)}}{{(\frac{1}{3})}^{i - 1} \cdot 1}$

Step 2.2.2.2.2

Cancel the common factor.

$r = \frac{{(\frac{1}{3})}^{i - 1} {(\frac{1}{3})}^{i + 0 - (i - 1)}}{{(\frac{1}{3})}^{i - 1} \cdot 1}$

Step 2.2.2.2.3

Rewrite the expression.

$r = \frac{{(\frac{1}{3})}^{i + 0 - (i - 1)}}{1}$

Step 2.2.2.2.4

Divide ${(\frac{1}{3})}^{i + 0 - (i - 1)}$ by $1$ .

$r = {(\frac{1}{3})}^{i + 0 - (i - 1)}$

Step 2.2.3

Add $i$ and $0$ .

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

Step 2.2.4

Simplify each term.

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Step 2.2.4.1

Apply the distributive property.

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

Step 2.2.4.2

Multiply $- 1$ by $- 1$ .

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

Step 2.2.5

Subtract $i$ from $i$ .

$r = {(\frac{1}{3})}^{0 + 1}$

Step 2.2.6

Add $0$ and $1$ .

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

Step 2.2.7

Simplify.

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

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 $\frac{1}{2} \cdot {(\frac{1}{3})}^{i - 1}$ .

$a = \frac{1}{2} \cdot {(\frac{1}{3})}^{1 - 1}$

Step 4.2

Simplify.

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Step 4.2.1

Subtract $1$ from $1$ .

$a = \frac{1}{2} \cdot {(\frac{1}{3})}^{0}$

Step 4.2.2

Apply the product rule to $\frac{1}{3}$ .

$a = \frac{1}{2} \cdot \frac{1^{0}}{3^{0}}$

Step 4.2.3

Combine.

$a = \frac{1 \cdot 1^{0}}{2 \cdot 3^{0}}$

Step 4.2.4

Multiply $1$ by $1^{0}$ by adding the exponents.

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Step 4.2.4.1

Multiply $1$ by $1^{0}$ .

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Step 4.2.4.1.1

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

$a = \frac{1^{1} \cdot 1^{0}}{2 \cdot 3^{0}}$

Step 4.2.4.1.2

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

$a = \frac{1^{1 + 0}}{2 \cdot 3^{0}}$

Step 4.2.4.2

Add $1$ and $0$ .

$a = \frac{1^{1}}{2 \cdot 3^{0}}$

Step 4.2.5

Simplify $1^{1}$ .

$a = \frac{1}{2 \cdot 3^{0}}$

Step 4.2.6

Anything raised to $0$ is $1$ .

$a = \frac{1}{2 \cdot 1}$

Step 4.2.7

Multiply $2$ by $1$ .

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

Step 5

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

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

Step 6

Simplify.

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Step 6.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 6.2

Simplify the denominator.

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Step 6.2.1

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

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

Step 6.2.2

Combine the numerators over the common denominator.

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

Step 6.2.3

Subtract $1$ from $3$ .

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

Step 6.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 6.4

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

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

Step 6.5

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

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Step 6.5.1

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

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

Step 6.5.2

Multiply $2$ by $2$ .

$\frac{3}{4}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{3}{4}$

Decimal Form:

$0.75$