Resources

Resources

Bytelearn - cat image with glasses

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

Find the sum of the infinite geometric series. $\newline$ $-1 - \frac{1}{4} - \frac{1}{16} - \frac{1}{64} +$ $\newline$ Write your answer as an integer or a fraction in simplest form. $\newline$ $\_\_$

View step-by-step help

View step-by-step help

Find the value of an infinite geometric series

Full solution

Q. Find the sum of the infinite geometric series.

\newline

-1 - \frac{1}{4} - \frac{1}{16} - \frac{1}{64} +

\newline

Write your answer as an integer or a fraction in simplest form.

\newline

\_\_

Identify Terms: Identify the first term $a$ and the common ratio $r$ of the geometric series. $\newline$ The first term $a$ is $-1$ , and the common ratio $r$ can be found by dividing the second term by the first term, which is $(-1/4) / (-1) = 1/4$ .
Check Common Ratio: Check if the common ratio's absolute value is less than $1$ to ensure the series converges. $\newline$ The common ratio $r$ is $\frac{1}{4}$ , and its absolute value $|\frac{1}{4}|$ is less than $1$ , which means the series converges.
Use Sum Formula: Use the formula for the sum of an infinite geometric series, which is $S = \frac{a}{1 - r}$ , where $S$ is the sum, $a$ is the first term, and $r$ is the common ratio. $\newline$ Substitute the values of $a$ and $r$ into the formula to find the sum. $\newline$ $S = \frac{-1}{1 - (\frac{1}{4})}$
Simplify Expression: Simplify the expression to find the sum. $\newline$ $S = \frac{-1}{1 - \frac{1}{4}}$ $\newline$ $S = \frac{-1}{\frac{3}{4}}$ $\newline$ $S = (-1) \times \frac{4}{3}$ $\newline$ $S = -\frac{4}{3}$

More problems from Find the value of an infinite geometric series

Question

Find the sum of the finite arithmetic series.

\sum_{n=1}^{10} (7n+4)

\newline

Posted 3 months ago

Question

Type the missing number in this sequence: `55,\59,\63,\text{_____},\71,\75,\79`

Posted 2 months ago

Question

Type the missing number in this sequence: `2, 4 8`, ______,`32`

Posted 2 months ago

Question

What kind of sequence is this?

2, 10, 50, 250, \ldots

Choices:Choices:

\newline

\text{[A]arithmetic}

\newline

\text{[B]geometric}

\newline

\text{[C]both}

\newline

\text{[D]neither}

Posted 2 months ago

Question

What is the missing number in this pattern?

1, 4, 9, 16, 25, 36, 49, 64, 81, \_\_\_\_

Posted 2 months ago

Question

Classify the series.

\sum_{n = 0}^{12} (n + 2)^3

\newline

\newline

\text{[A]arithmetic}

\newline

\text{[B]geometric}

\newline

\text{[C]both}

\newline

\text{[D]neither}

Posted 2 months ago

Question

Find the first three partial sums of the series.

\newline

1 + 6 + 11 + 16 + 21 + 26 + \cdots

\newline

Write your answers as integers or fractions in simplest form.

\newline

S_1 =

\newline

S_2 =

\newline

S_3 =

Posted 2 months ago

Question

Find the third partial sum of the series.

\newline

3 + 9 + 15 + 21 + 27 + 33 + \cdots

\newline

Write your answer as an integer or a fraction in simplest form.

\newline

S_3 =

Posted 2 months ago

Question

Find the first three partial sums of the series.

\newline

1 + 7 + 13 + 19 + 25 + 31 + \cdots

\newline

Write your answers as integers or fractions in simplest form.

\newline

S_1 =

\newline

S_2 =

\newline

S_3 =

Posted 3 months ago

Question

Does the infinite geometric series converge or diverge?

\newline

1 + \frac{3}{4} + \frac{9}{16} + \frac{27}{64} + \cdots

\newline

\newline

\text{[A]converge}

\newline

\text{[B]diverge}

Posted 2 months ago