Calculus Examples

\infty \sum n = 1 {(\frac{2}{3})}^{n}

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

Step 1

The sum of an infinite geometric series can be found using the formula

\frac{a}{1 - r}

$\frac{a}{1 - r}$ where

a

$a$ is the first term and

r

$r$ 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}}

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

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Step 2.1

Substitute

a_{n}

$a_{n}$ and

a_{n + 1}

$a_{n + 1}$ into the formula for

r

$r$ .

r = \frac{{(\frac{2}{3})}^{n + 1}}{{(\frac{2}{3})}^{n}}

$r = \frac{{(\frac{2}{3})}^{n + 1}}{{(\frac{2}{3})}^{n}}$

Step 2.2

Cancel the common factor of

{(\frac{2}{3})}^{n + 1}

${(\frac{2}{3})}^{n + 1}$ and

{(\frac{2}{3})}^{n}

${(\frac{2}{3})}^{n}$ .

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Step 2.2.1

Factor

{(\frac{2}{3})}^{n}

${(\frac{2}{3})}^{n}$ out of

{(\frac{2}{3})}^{n + 1}

${(\frac{2}{3})}^{n + 1}$ .

r = \frac{{(\frac{2}{3})}^{n} \frac{2}{3}}{{(\frac{2}{3})}^{n}}

$r = \frac{{(\frac{2}{3})}^{n} \frac{2}{3}}{{(\frac{2}{3})}^{n}}$

Step 2.2.2

Cancel the common factors.

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Step 2.2.2.1

Multiply by

1

$1$ .

r = \frac{{(\frac{2}{3})}^{n} \frac{2}{3}}{{(\frac{2}{3})}^{n} \cdot 1}

$r = \frac{{(\frac{2}{3})}^{n} \frac{2}{3}}{{(\frac{2}{3})}^{n} \cdot 1}$

Step 2.2.2.2

Cancel the common factor.

$r = \frac{{(\frac{2}{3})}^{n} \frac{2}{3}}{{(\frac{2}{3})}^{n} \cdot 1}$

Step 2.2.2.3

Rewrite the expression.

$r = \frac{\frac{2}{3}}{1}$

Step 2.2.2.4

Divide

$\frac{2}{3}$ by

$1$ .

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

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

$n$ into

${(\frac{2}{3})}^{n}$ .

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

Step 4.2

Simplify.

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

Step 5

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

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

Step 6

Simplify.

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Step 6.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 6.2

Simplify the denominator.

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Step 6.2.1

Write

$1$ as a fraction with a common denominator.

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

Step 6.2.2

Combine the numerators over the common denominator.

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

Step 6.2.3

Subtract

$2$ from

$3$ .

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

Step 6.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 6.4

Cancel the common factor of

$3$ .

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Step 6.4.1

Factor

$3$ out of

$1 \cdot 3$ .

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

Step 6.4.2

Cancel the common factor.

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

Step 6.4.3

Rewrite the expression.

$2$