# Root Test for series

• Nov 19th 2013, 03:16 PM
Dfaulk044
Root Test for series
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
• Nov 19th 2013, 04:14 PM
SlipEternal
Re: Root Test for series
I can't read your post. What is in the numerator, what is in the denominator, what is being taken to the n-th power? Use more parentheses.
• Nov 19th 2013, 04:37 PM
HallsofIvy
Re: Root Test for series
Quote:

Originally Posted by Dfaulk044
Using the root test or otherwise, determine whether the given series
converges or diverges.

A) ∑ (n^2+1/2n^2+n)^n

I think you mean $\sum_{n=1}^\infty \left(\frac{n^2+ 1}{2n^2+ n}\right)^n$
(what you wrote was a sum of $n^2+ (1/n^2)+n$)
Since you mention the root test, the nth root of each term is $\frac{n^2+ 1}{2n^2+ n}$. What is the limit of that as n goes to infinity? What does that tell you about the sum?

Quote:

n=1

∞B) ∑ (1-2/n)^n^2

n=1

Here, I assume you mean $\sum_{n= 1}^\infty \left(1- \frac{2}{n}\right)^{n^2}$
The nth root of that is $\left(1- \frac{2}{n}\right)^n$. What is the limit of that? (You really need to know that $\lim_{n\to\infty}\left(1+ \frac{1}{n}\right)^n= e$.)
• Nov 20th 2013, 06:20 AM
SlipEternal
Re: Root Test for series
Also helpful is the fact that $\lim_{n \to \infty} \left( 1 + \dfrac{x}{n} \right)^n = e^x$.