Respuesta :
Answer:
Given series is divergent
Step-by-step explanation:
Step(i):-
By using Ratio test
[tex]\lim_{n \to \infty} |\frac{a_{n+1} }{a_{n} } | = l[/tex]
a) [tex]\lim_{n \to \infty} |\frac{a_{n+1} }{a_{n} } | = l[/tex]
'l' is finite then the given ∑aₙ is convergent
b) [tex]\lim_{n \to \infty} |\frac{a_{n+1} }{a_{n} } | = l[/tex]
Here 'l' is infinite then the ∑aₙ is divergent
Step(ii):-
Given aₙ = [tex]\frac{n!}{n}[/tex]
[tex]a_{n+1} = \frac{(n+1)!}{n+1}[/tex]
[tex]\lim_{n \to \infty} |\frac{a_{n+1} }{a_{n} } | = \lim_{n \to \infty} |\frac{\frac{(n+1)!}{n+1} }{\frac{n!}{n} } |[/tex]
we know that n ! = n (n-1) (n-2) ......3.2.1
and also (n+1) ! = (n+1)n!
[tex]\lim_{n \to \infty} |\frac{a_{n+1} }{a_{n} } | = \lim_{n \to \infty} |\frac{\frac{n+1)n!}{n+1} }{\frac{n!}{n} } |[/tex]
[tex]\lim_{n \to \infty} |\frac{a_{n+1} }{a_{n} } | = \lim_{n \to \infty} n[/tex]
= ∞
Given sum of the series is divergent
