Calculus Examples

$\sum_{n = 1}^{\infty} - 4 {(- \frac{1}{2})}^{n - 1}$

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

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Step 2.1

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

$r = \frac{- 4 {(- \frac{1}{2})}^{n + 1 - 1}}{- 4 {(- \frac{1}{2})}^{n - 1}}$

Step 2.2

Simplify.

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Step 2.2.1

Cancel the common factor of $- 4$ .

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Step 2.2.1.1

Cancel the common factor.

$r = \frac{- 4 {(- \frac{1}{2})}^{n + 1 - 1}}{- 4 {(- \frac{1}{2})}^{n - 1}}$

Step 2.2.1.2

Rewrite the expression.

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

Step 2.2.2

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

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Step 2.2.2.1

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

$r = \frac{{(- \frac{1}{2})}^{n - 1} {(- \frac{1}{2})}^{n + 0 - (n - 1)}}{{(- \frac{1}{2})}^{n - 1}}$

Step 2.2.2.2

Cancel the common factors.

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Step 2.2.2.2.1

Multiply by $1$ .

$r = \frac{{(- \frac{1}{2})}^{n - 1} {(- \frac{1}{2})}^{n + 0 - (n - 1)}}{{(- \frac{1}{2})}^{n - 1} \cdot 1}$

Step 2.2.2.2.2

Cancel the common factor.

$r = \frac{{(- \frac{1}{2})}^{n - 1} {(- \frac{1}{2})}^{n + 0 - (n - 1)}}{{(- \frac{1}{2})}^{n - 1} \cdot 1}$

Step 2.2.2.2.3

Rewrite the expression.

$r = \frac{{(- \frac{1}{2})}^{n + 0 - (n - 1)}}{1}$

Step 2.2.2.2.4

Divide ${(- \frac{1}{2})}^{n + 0 - (n - 1)}$ by $1$ .

$r = {(- \frac{1}{2})}^{n + 0 - (n - 1)}$

Step 2.2.3

Add $n$ and $0$ .

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

Step 2.2.4

Simplify each term.

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Step 2.2.4.1

Apply the distributive property.

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

Step 2.2.4.2

Multiply $- 1$ by $- 1$ .

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

Step 2.2.5

Subtract $n$ from $n$ .

$r = {(- \frac{1}{2})}^{0 + 1}$

Step 2.2.6

Add $0$ and $1$ .

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

Step 2.2.7

Simplify.

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

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 $1$ for $n$ into $- 4 {(- \frac{1}{2})}^{n - 1}$ .

$a = - 4 {(- \frac{1}{2})}^{1 - 1}$

Step 4.2

Simplify.

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Step 4.2.1

Subtract $1$ from $1$ .

$a = - 4 {(- \frac{1}{2})}^{0}$

Step 4.2.2

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 4.2.2.1

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

$a = - 4 ({(- 1)}^{0} {(\frac{1}{2})}^{0})$

Step 4.2.2.2

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

$a = - 4 ({(- 1)}^{0} \frac{1^{0}}{2^{0}})$

Step 4.2.3

Anything raised to $0$ is $1$ .

$a = - 4 (1 \frac{1^{0}}{2^{0}})$

Step 4.2.4

Multiply $\frac{1^{0}}{2^{0}}$ by $1$ .

$a = - 4 \frac{1^{0}}{2^{0}}$

Step 4.2.5

Anything raised to $0$ is $1$ .

$a = - 4 \frac{1}{2^{0}}$

Step 4.2.6

Anything raised to $0$ is $1$ .

$a = - 4 (\frac{1}{1})$

Step 4.2.7

Cancel the common factor of $1$ .

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Step 4.2.7.1

Cancel the common factor.

$a = - 4 (\frac{1}{1})$

Step 4.2.7.2

Rewrite the expression.

$a = - 4 \cdot 1$

Step 4.2.8

Multiply $- 4$ by $1$ .

$a = - 4$

Step 5

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

$\frac{- 4}{1 - - \frac{1}{2}}$

Step 6

Simplify.

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Step 6.1

Simplify the denominator.

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Step 6.1.1

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

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Step 6.1.1.1

Multiply $- 1$ by $- 1$ .

$\frac{- 4}{1 + 1 (\frac{1}{2})}$

Step 6.1.1.2

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

$\frac{- 4}{1 + \frac{1}{2}}$

Step 6.1.2

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

$\frac{- 4}{\frac{2}{2} + \frac{1}{2}}$

Step 6.1.3

Combine the numerators over the common denominator.

$\frac{- 4}{\frac{2 + 1}{2}}$

Step 6.1.4

Add $2$ and $1$ .

$\frac{- 4}{\frac{3}{2}}$

Step 6.2

Multiply the numerator by the reciprocal of the denominator.

$- 4 (\frac{2}{3})$

Step 6.3

Multiply $- 4 (\frac{2}{3})$ .

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Step 6.3.1

Combine $- 4$ and $\frac{2}{3}$ .

$\frac{- 4 \cdot 2}{3}$

Step 6.3.2

Multiply $- 4$ by $2$ .

$\frac{- 8}{3}$

Step 6.4

Move the negative in front of the fraction.

$- \frac{8}{3}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$- \frac{8}{3}$

Decimal Form:

$- 2.\overline{6}$

Mixed Number Form:

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