Precalculus Examples

$1 - \frac{2}{3} + \frac{4}{9} - \frac{8}{27} + \dots$

Step 1

This is a geometric sequence since there is a common ratio between each term. In this case, multiplying the previous term in the sequence by $- \frac{2}{3}$ gives the next term. In other words, $a_{n} = a_{1} r^{n - 1}$ .

Geometric Sequence: $r = - \frac{2}{3}$

Step 2

The sum of a series $S_{n}$ is calculated using the formula $S_{n} = \frac{a (1 - r^{n})}{1 - r}$ . For the sum of an infinite geometric series $S_{\infty}$ , as $n$ approaches $\infty$ , $1 - r^{n}$ approaches $1$ . Thus, $\frac{a (1 - r^{n})}{1 - r}$ approaches $\frac{a}{1 - r}$ .

$S_{\infty} = \frac{a}{1 - r}$

Step 3

The values $a = 1$ and $r = - \frac{2}{3}$ can be put in the equation $S_{\infty}$ .

$S_{\infty} = \frac{1}{1 + \frac{2}{3}}$

Step 4

Simplify the equation to find $S_{\infty}$ .

Tap for more steps...

Step 4.1

Simplify the denominator.

Tap for more steps...

Step 4.1.1

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

$S_{\infty} = \frac{1}{\frac{3}{3} + \frac{2}{3}}$

Step 4.1.2

Combine the numerators over the common denominator.

$S_{\infty} = \frac{1}{\frac{3 + 2}{3}}$

Step 4.1.3

Add $3$ and $2$ .

$S_{\infty} = \frac{1}{\frac{5}{3}}$

Step 4.2

Multiply the numerator by the reciprocal of the denominator.

$S_{\infty} = 1 (\frac{3}{5})$

Step 4.3

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

$S_{\infty} = \frac{3}{5}$