Answer:
This series is divergent
B.
Step-by-step explanation:
We are given a series
Firstly, we will find nth term
Numerator:
2 , 4, 8 , 16 , .....
[tex]a_n=2^n[/tex]
Denominator:
1^2 , 2^2 , 3^2 , ....
[tex]b_n=n^2[/tex]
now, we can find nth term
[tex]c_n=\frac{2^n}{n^2}[/tex]
We can use ratio test
[tex]L= \lim_{n \to \infty} c_n= \lim_{n \to \infty} \frac{\frac{2^{n+1}}{(n+1)^2}}{\frac{2^n}{n^2}}[/tex]
[tex]L=\lim _{n\to \infty \:}\left(\frac{2n^2}{\left(n+1\right)^2}\right)[/tex]
[tex]L=2\left(\lim _{n\to \infty \:}\left(\frac{1}{1+\frac{1}{n}}\right)\right)^2[/tex]
[tex]L=2\left(\frac{1}{1}\right)^2[/tex]
[tex]L=2>1[/tex]
Since, it is greater than 1
so, this series is divergent
