Calculus Examples

$\sum_{k = 1}^{\infty} {(\frac{2}{5})}^{k}$

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 $r$ .

$r = \frac{{(\frac{2}{5})}^{k + 1}}{{(\frac{2}{5})}^{k}}$

Cancel the common factor of ${(\frac{2}{5})}^{k + 1}$ and ${(\frac{2}{5})}^{k}$ .

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Factor ${(\frac{2}{5})}^{k}$ out of ${(\frac{2}{5})}^{k + 1}$ .

$r = \frac{{(\frac{2}{5})}^{k} \frac{2}{5}}{{(\frac{2}{5})}^{k}}$

Cancel the common factors.

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Multiply by $1$ .

$r = \frac{{(\frac{2}{5})}^{k} \frac{2}{5}}{{(\frac{2}{5})}^{k} \cdot 1}$

Cancel the common factor.

$r = \frac{{(\frac{2}{5})}^{k} \frac{2}{5}}{{(\frac{2}{5})}^{k} \cdot 1}$

Rewrite the expression.

$r = \frac{\frac{2}{5}}{1}$

Divide $\frac{2}{5}$ by $1$ .

$r = \frac{2}{5}$

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 ${(\frac{2}{5})}^{k}$ .

$a = {(\frac{2}{5})}^{1}$

Simplify.

$a = \frac{2}{5}$

Step 5

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

$\frac{\frac{2}{5}}{1 - \frac{2}{5}}$

Step 6

Simplify.

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Multiply the numerator by the reciprocal of the denominator.

$\frac{2}{5} \cdot \frac{1}{1 - \frac{2}{5}}$

Simplify the denominator.

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

$\frac{2}{5} \cdot \frac{1}{\frac{5}{5} - \frac{2}{5}}$

Combine the numerators over the common denominator.

$\frac{2}{5} \cdot \frac{1}{\frac{5 - 2}{5}}$

Subtract $2$ from $5$ .

$\frac{2}{5} \cdot \frac{1}{\frac{3}{5}}$

Multiply the numerator by the reciprocal of the denominator.

$\frac{2}{5} (1 (\frac{5}{3}))$

Multiply $\frac{5}{3}$ by $1$ .

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

Cancel the common factor of $5$ .

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Cancel the common factor.

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

Rewrite the expression.

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

Combine $2$ and $\frac{1}{3}$ .

$\frac{2}{3}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{2}{3}$

Decimal Form:

$0.\overline{6}$