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

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- Nov 19th 2013, 03:16 PMDfaulk044Root 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 PMSlipEternalRe: 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 PMHallsofIvyRe: Root Test for series
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?

Quote:

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 $\displaystyle \lim_{n\to\infty}\left(1+ \frac{1}{n}\right)^n= e$.) - Nov 20th 2013, 06:20 AMSlipEternalRe: Root Test for series
Also helpful is the fact that $\displaystyle \lim_{n \to \infty} \left( 1 + \dfrac{x}{n} \right)^n = e^x$.