Calculus Examples

$\sum_{n = 0}^{\infty} {(\frac{1}{2})}^{n}$

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 $r$ .

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

Step 2.2

Cancel the common factor of ${(\frac{1}{2})}^{n + 1}$ and ${(\frac{1}{2})}^{n}$ .

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Step 2.2.1

Factor ${(\frac{1}{2})}^{n}$ out of ${(\frac{1}{2})}^{n + 1}$ .

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

Step 2.2.2

Cancel the common factors.

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Step 2.2.2.1

Multiply by $1$ .

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

Step 2.2.2.2

Cancel the common factor.

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

Step 2.2.2.3

Rewrite the expression.

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

Step 2.2.2.4

Divide $\frac{1}{2}$ by $1$ .

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

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 $0$ for $n$ into ${(\frac{1}{2})}^{n}$ .

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

Step 4.2

Simplify.

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Step 4.2.1

Apply the product rule to $\frac{1}{2}$ .

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

Step 4.2.2

Anything raised to $0$ is $1$ .

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

Step 4.2.3

Anything raised to $0$ is $1$ .

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

Step 4.2.4

Divide $1$ by $1$ .

$a = 1$

Step 5

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

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

Step 6

Simplify.

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Step 6.1

Simplify the denominator.

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Step 6.1.1

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

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

Step 6.1.2

Combine the numerators over the common denominator.

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

Step 6.1.3

Subtract $1$ from $2$ .

$\frac{1}{\frac{1}{2}}$

Step 6.2

Multiply the numerator by the reciprocal of the denominator.

$1 \cdot 2$

Step 6.3

Multiply $2$ by $1$ .

$2$