Calculus Examples

$\sum_{i = 1}^{\infty} 3 {(\frac{3}{10})}^{i - 1}$

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.

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

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

$r = \frac{3 {(\frac{3}{10})}^{i + 1 - 1}}{3 {(\frac{3}{10})}^{i - 1}}$

Simplify.

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Cancel the common factor of $3$ .

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Cancel the common factor.

$r = \frac{3 {(\frac{3}{10})}^{i + 1 - 1}}{3 {(\frac{3}{10})}^{i - 1}}$

Rewrite the expression.

$r = \frac{{(\frac{3}{10})}^{i + 1 - 1}}{{(\frac{3}{10})}^{i - 1}}$

Cancel the common factor of ${(\frac{3}{10})}^{i + 1 - 1}$ and ${(\frac{3}{10})}^{i - 1}$ .

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Factor ${(\frac{3}{10})}^{i - 1}$ out of ${(\frac{3}{10})}^{i + 1 - 1}$ .

$r = \frac{{(\frac{3}{10})}^{i - 1} {(\frac{3}{10})}^{i + 0 - (i - 1)}}{{(\frac{3}{10})}^{i - 1}}$

Cancel the common factors.

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Multiply by $1$ .

$r = \frac{{(\frac{3}{10})}^{i - 1} {(\frac{3}{10})}^{i + 0 - (i - 1)}}{{(\frac{3}{10})}^{i - 1} \cdot 1}$

Cancel the common factor.

$r = \frac{{(\frac{3}{10})}^{i - 1} {(\frac{3}{10})}^{i + 0 - (i - 1)}}{{(\frac{3}{10})}^{i - 1} \cdot 1}$

Rewrite the expression.

$r = \frac{{(\frac{3}{10})}^{i + 0 - (i - 1)}}{1}$

Divide ${(\frac{3}{10})}^{i + 0 - (i - 1)}$ by $1$ .

$r = {(\frac{3}{10})}^{i + 0 - (i - 1)}$

Simplify each term.

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Apply the distributive property.

$r = {(\frac{3}{10})}^{i + 0 - i - - 1}$

Multiply $- 1$ by $- 1$ .

$r = {(\frac{3}{10})}^{i + 0 - i + 1}$

Add $i$ and $0$ .

$r = {(\frac{3}{10})}^{i - i + 1}$

Subtract $i$ from $i$ .

$r = {(\frac{3}{10})}^{0 + 1}$

Add $0$ and $1$ .

$r = {(\frac{3}{10})}^{1}$

Simplify.

$r = \frac{3}{10}$

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

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Substitute $1$ for $i$ into $3 {(\frac{3}{10})}^{i - 1}$ .

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

Simplify.

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Subtract $1$ from $1$ .

$a = 3 {(\frac{3}{10})}^{0}$

Apply the product rule to $\frac{3}{10}$ .

$a = 3 \frac{3^{0}}{10^{0}}$

Anything raised to $0$ is $1$ .

$a = 3 \frac{1}{10^{0}}$

Anything raised to $0$ is $1$ .

$a = 3 (\frac{1}{1})$

Cancel the common factor of $1$ .

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Cancel the common factor.

$a = 3 (\frac{1}{1})$

Rewrite the expression.

$a = 3 \cdot 1$

Multiply $3$ by $1$ .

$a = 3$

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

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

Simplify.

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Simplify the denominator.

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

$\frac{3}{\frac{10}{10} - \frac{3}{10}}$

Combine the numerators over the common denominator.

$\frac{3}{\frac{10 - 3}{10}}$

Subtract $3$ from $10$ .

$\frac{3}{\frac{7}{10}}$

Multiply the numerator by the reciprocal of the denominator.

$3 (\frac{10}{7})$

Multiply $3 (\frac{10}{7})$ .

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Combine $3$ and $\frac{10}{7}$ .

$\frac{3 \cdot 10}{7}$

Multiply $3$ by $10$ .

$\frac{30}{7}$

The result can be shown in multiple forms.

Exact Form:

$\frac{30}{7}$

Decimal Form:

$4.\overline{285714}$

Mixed Number Form:

$4 \frac{2}{7}$