karush said:Calculate the sum for the infinite geometric series
$4+2+1+\frac{1}{2}+...$
all I know is the ratio is $\frac{1}{2}$
$\displaystyle\sum_{n}^{\infty}a{r}^{n}$
assume this is used
karush said:Calculate the sum for the infinite geometric series
$4+2+1+\frac{1}{2}+...$
$a=5 \ \ r=\frac{1}{2}$
$\displaystyle\sum_{n= 0}^{\infty}a{r}^{n}= a (\frac{1}{1-r})
= 4 (\frac{1}{1-\frac{1}{2}})= 4 * 2 = 8$
An infinite geometric series is a series of numbers that continues on infinitely, with each number being multiplied by a common ratio. It can be represented as a + ar + ar^2 + ar^3 + ..., where a is the first term and r is the common ratio.
The sum of an infinite geometric series can be calculated using the formula a/(1-r), where a is the first term and r is the common ratio. This formula only works if the absolute value of r is less than 1, otherwise the series will not converge.
Yes, an infinite geometric series can have a finite sum if the absolute value of the common ratio is less than 1. If the absolute value is greater than or equal to 1, the series will not converge and will have an infinite sum.
A finite geometric series has a finite number of terms, while an infinite geometric series has an infinite number of terms. A finite geometric series can have a finite or infinite sum depending on the value of the common ratio, while an infinite geometric series will always have a finite sum if the common ratio is less than 1.
Infinite geometric series are commonly used in finance and economics to calculate compound interest, where the common ratio represents the interest rate. They are also used in physics and engineering to model exponential growth and decay processes. Additionally, infinite geometric series are used in computer algorithms and data compression techniques.