Using the root test or otherwise, determine whether the given series
converges or diverges.
∞
A) ∑ (n^2+1/2n^2+n)^n
n=1
∞B) ∑ (1-2/n)^n^2
n=1
I think you mean $\displaystyle \sum_{n=1}^\infty \left(\frac{n^2+ 1}{2n^2+ n}\right)^n$
(what you wrote was a sum of $\displaystyle n^2+ (1/n^2)+n$)
Since you mention the root test, the nth root of each term is $\displaystyle \frac{n^2+ 1}{2n^2+ n}$. What is the limit of that as n goes to infinity? What does that tell you about the sum?
Here, I assume you mean $\displaystyle \sum_{n= 1}^\infty \left(1- \frac{2}{n}\right)^{n^2}$n=1
∞B) ∑ (1-2/n)^n^2
n=1
The nth root of that is $\displaystyle \left(1- \frac{2}{n}\right)^n$. What is the limit of that? (You really need to know that $\displaystyle \lim_{n\to\infty}\left(1+ \frac{1}{n}\right)^n= e$.)