Calculus Examples

$\sum_{k = 1}^{\infty} 6 {(\frac{1}{5})}^{k - 1}$

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

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

$r = \frac{6 {(\frac{1}{5})}^{k + 1 - 1}}{6 {(\frac{1}{5})}^{k - 1}}$

Simplify.

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Cancel the common factor of $6$ .

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Cancel the common factor.

$r = \frac{6 {(\frac{1}{5})}^{k + 1 - 1}}{6 {(\frac{1}{5})}^{k - 1}}$

Rewrite the expression.

$r = \frac{{(\frac{1}{5})}^{k + 1 - 1}}{{(\frac{1}{5})}^{k - 1}}$

Cancel the common factor of ${(\frac{1}{5})}^{k + 1 - 1}$ and ${(\frac{1}{5})}^{k - 1}$ .

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Factor ${(\frac{1}{5})}^{k - 1}$ out of ${(\frac{1}{5})}^{k + 1 - 1}$ .

$r = \frac{{(\frac{1}{5})}^{k - 1} {(\frac{1}{5})}^{k + 0 - (k - 1)}}{{(\frac{1}{5})}^{k - 1}}$

Cancel the common factors.

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Multiply by $1$ .

$r = \frac{{(\frac{1}{5})}^{k - 1} {(\frac{1}{5})}^{k + 0 - (k - 1)}}{{(\frac{1}{5})}^{k - 1} \cdot 1}$

Cancel the common factor.

$r = \frac{{(\frac{1}{5})}^{k - 1} {(\frac{1}{5})}^{k + 0 - (k - 1)}}{{(\frac{1}{5})}^{k - 1} \cdot 1}$

Rewrite the expression.

$r = \frac{{(\frac{1}{5})}^{k + 0 - (k - 1)}}{1}$

Divide ${(\frac{1}{5})}^{k + 0 - (k - 1)}$ by $1$ .

$r = {(\frac{1}{5})}^{k + 0 - (k - 1)}$

Add $k$ and $0$ .

$r = {(\frac{1}{5})}^{k - (k - 1)}$

Simplify each term.

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Apply the distributive property.

$r = {(\frac{1}{5})}^{k - k - - 1}$

Multiply $- 1$ by $- 1$ .

$r = {(\frac{1}{5})}^{k - k + 1}$

Subtract $k$ from $k$ .

$r = {(\frac{1}{5})}^{0 + 1}$

Add $0$ and $1$ .

$r = {(\frac{1}{5})}^{1}$

Simplify.

$r = \frac{1}{5}$

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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Substitute $1$ for $k$ into $6 {(\frac{1}{5})}^{k - 1}$ .

$a = 6 {(\frac{1}{5})}^{1 - 1}$

Simplify.

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Subtract $1$ from $1$ .

$a = 6 {(\frac{1}{5})}^{0}$

Apply the product rule to $\frac{1}{5}$ .

$a = 6 \frac{1^{0}}{5^{0}}$

Anything raised to $0$ is $1$ .

$a = 6 \frac{1}{5^{0}}$

Anything raised to $0$ is $1$ .

$a = 6 (\frac{1}{1})$

Cancel the common factor of $1$ .

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Cancel the common factor.

$a = 6 (\frac{1}{1})$

Rewrite the expression.

$a = 6 \cdot 1$

Multiply $6$ by $1$ .

$a = 6$

Step 5

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

$\frac{6}{1 - \frac{1}{5}}$

Step 6

Simplify.

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Simplify the denominator.

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

$\frac{6}{\frac{5}{5} - \frac{1}{5}}$

Combine the numerators over the common denominator.

$\frac{6}{\frac{5 - 1}{5}}$

Subtract $1$ from $5$ .

$\frac{6}{\frac{4}{5}}$

Multiply the numerator by the reciprocal of the denominator.

$6 (\frac{5}{4})$

Cancel the common factor of $2$ .

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Factor $2$ out of $6$ .

$2 (3) \frac{5}{4}$

Factor $2$ out of $4$ .

$2 \cdot 3 \frac{5}{2 \cdot 2}$

Cancel the common factor.

$2 \cdot 3 \frac{5}{2 \cdot 2}$

Rewrite the expression.

$3 (\frac{5}{2})$

Combine $3$ and $\frac{5}{2}$ .

$\frac{3 \cdot 5}{2}$

Multiply $3$ by $5$ .

$\frac{15}{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{15}{2}$

Decimal Form:

$7.5$

Mixed Number Form:

$7 \frac{1}{2}$