Answer:
This series is divergent
F
Step-by-step explanation:
we are given a series
Firstly, we will find nth term
Numerator:
3, 4, 5,...
so, nth term will be
[tex]a_n=n+3[/tex]
Denominator:
4,5,6,....
so, nth term will be
[tex]b_n=n+4[/tex]
so, we can find it's nth term as
[tex]c_n=\frac{n+3}{n+4}[/tex]
we can use divergent test
[tex]\lim_{n \to \infty} c_n=\lim_{n \to \infty} \frac{n+3}{n+4}[/tex]
we can divide top and bottom by n
[tex]\lim_{n \to \infty} c_n= \lim_{n \to \infty} \frac{n/n+3/n}{n/n+4/n}[/tex]
[tex]\lim_{n \to \infty} c_n= \lim_{n \to \infty} \frac{1+3/n}{1+4/n}[/tex]
now, we can plug n=inf
[tex]\lim_{n \to \infty} c_n= \frac{1+0}{1+0}[/tex]
[tex]\lim_{n \to \infty} c_n=1[/tex]
Since, it is non-zero value
so, this series is divergent
