Which of the following series is divergent?
a) 1+3(1/4)+9(1/4)^2+27(1/4)^3...
b) 1+3(1/5)+9(1/5)^2+27(1/5)^3...
c) 1+3(1/7)+9(1/7)^2+27(1/7)^3...
d) 1+3(1/2)+9(1/2)^2+27(1/2)^3...
How do you determine if a series in convergent or divergent???
The book that I have is about as clear as mud and I do not understand...
Thanks for your help!!
This would be a lengthy reply, so see if you can make sense out of this
http://faculty.msmary.edu/heinold/m248spring2007/m248spring2007_convergence_tests.pdf
I will take a look at the first question
1+3(1/4)+9(1/4)^2+27(1/4)^3...
= 1 + 3^1 / 4^1 + 3^2 / 4^2 + ..
= 1 + summation of (3/4)^n as n goes from 1 to infinitity
Now compare this result with what it says about the summation of a geometric series at the top of the link I gave you.
good luck.
To determine if a series is convergent or divergent, you can use several different methods, including the comparison test, the ratio test, and the root test.
Let's go through each option and see if we can determine if it is convergent or divergent:
a) 1+3(1/4)+9(1/4)^2+27(1/4)^3...
For this series, we can see that each term is a multiple of a power of (1/4). We can use the ratio test to determine convergence. The ratio test states that if the absolute value of the ratio of consecutive terms approaches a value less than 1 as n approaches infinity, the series is convergent.
Let's calculate the ratio of consecutive terms:
(3/4) / (1/4) = 3
The limit of this ratio as n approaches infinity is 3, which is greater than 1. Therefore, the series is divergent.
b) 1+3(1/5)+9(1/5)^2+27(1/5)^3...
To check this series, we can again use the ratio test:
(3/5) / (1/5)= 3
The limit of this ratio is also 3, which is greater than 1. Therefore, this series is also divergent.
c) 1+3(1/7)+9(1/7)^2+27(1/7)^3...
Using the ratio test again:
(3/7) / (1/7) = 3
The ratio is 3, which is greater than 1. Thus, this series is divergent as well.
d) 1+3(1/2)+9(1/2)^2+27(1/2)^3...
Applying the ratio test once more:
(3/2) / (1/2) = 3
The ratio is 3, which is greater than 1. Therefore, this series is divergent too.
In summary, all the given series (a, b, c, d) are divergent because the ratio of consecutive terms is always equal to 3, which is greater than 1.