Do these converge absolutely, conditionally, or diverge?
sigma from n = 1 to infinity of (n + 1)^2 / n!
sigma from n = 1 to infinity of (1/ (arctan(n))^(2n)
explain please!
Do these converge absolutely, conditionally, or diverge?
sigma from n = 1 to infinity of (n + 1)^2 / n!
sigma from n = 1 to infinity of (1/ (arctan(n))^(2n)
explain please!
For the first, try the ratio test.
Evaluate $\displaystyle \displaystyle \lim_{n \to \infty}\left|\frac{a_{n+1}}{a_n}\right|$. If this is $\displaystyle \displaystyle <1$ the series converges absolutely, if this is $\displaystyle \displaystyle >1$ the series diverges. If this $\displaystyle \displaystyle =1$ the test is inconclusive.
so for the first one I applied the ratio test and got
lim (as n goes to infinity) of (n^2 + 2n + 1) / ( (n+1)(n^2) ) + lim (as n goes to infinity) (2n + 1)/( (n+1)(n^2) + lim (as n goes to infinity) of 1/(n+1)
and since all of these go to zero then the sum converges absolutely?
and for the second one you have
sum (from 1 to infinity) 1 / (pi/2)^2n
and because pi/2 is greater than 1 it just diverges by geometric series test?
You are mixing up two separate suggestions. I thought you wanted to solve the series but you just want to know if they converge. Just follow Post #3
$\displaystyle \displaystyle a_{n+1}=\frac{(n+2)^2}{(n+1)!}$
$\displaystyle \displaystyle \frac{\frac{(n+2)^2}{(n+1)!}}{\frac{(n+1)^2}{n!}}\ rightarrow \frac{(n+2)^2}{(n+1)^3}\rightarrow \frac{n^2}{n^3}\rightarrow\lim_{n\to\infty}\frac{1 }{n}=0$