IDNLearn.com connects you with a community of knowledgeable individuals ready to help. Our experts provide timely, comprehensive responses to ensure you have the information you need.
Sagot :
[tex]1/2+1/4+1/8+1/16+...=\sum\limits_{n=1}^\infty (\frac{1}{2})^n = \frac{\frac{1}{2}}{1-\frac{1}{2}}=1[/tex]
We can see that consecutive fractions are made from 1/2 to consecutive powers. Because we begin with 1/2, n=1
We will infinitely add fractions , hence Lemniscate sign.
In other words you cannot find a number which needs to be added to the geometric series to get "1", therefore the answer is 1. I remember the teacher explaining it this way :)
We can see that consecutive fractions are made from 1/2 to consecutive powers. Because we begin with 1/2, n=1
We will infinitely add fractions , hence Lemniscate sign.
In other words you cannot find a number which needs to be added to the geometric series to get "1", therefore the answer is 1. I remember the teacher explaining it this way :)
Answer:
The sum of the given geometric series is, 1
Step-by-step explanation:
Geometric sequence states that a sequence of numbers that follows a pattern were the next term is found by multiplying by a constant called the common ratio (r).
The sum of the infinite terms of a geometric series is given by:
[tex]S_\infty = \frac{a}{1-r}[/tex] ......[1] ;where [tex]0<r<1[/tex]
Given the series: [tex]\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+.....[/tex]
Since, this series is geometric series with constant term(r) = [tex]\frac{1}{2}[/tex]
Since,
[tex]\frac{\frac{1}{4}}{\frac{1}{2} } =\frac{2}{4} = \frac{1}{2}[/tex],
[tex]\frac{\frac{1}{8}}{\frac{1}{4}} =\frac{4}{8} = \frac{1}{2}[/tex] and so on....
Here, first term(a) = [tex]\frac{1}{2}[/tex]
Substitute the values of a and r in [1] we get;
[tex]S_\infty = \frac{\frac{1}{2}}{1-\frac{1}{2}}[/tex] where r = [tex]\frac{1}{2}< 1[/tex]
[tex]S_\infty = \frac{\frac{1}{2}}{\frac{2-1}{2}}[/tex]
or
[tex]S_\infty = \frac{\frac{1}{2}}{\frac{1}{2}}[/tex]
Simplify:
[tex]S_\infty = 1[/tex]
Therefore, the sum of the infinite geometric series is, 1
Thank you for joining our conversation. Don't hesitate to return anytime to find answers to your questions. Let's continue sharing knowledge and experiences! Thank you for choosing IDNLearn.com. We’re dedicated to providing clear answers, so visit us again for more solutions.