Infinite Series (The Ratio Test)

[Fernando Rios]

Homework Statement

Use the ratio test to find whether the following series converge or diverge:

Relevant Equations

∑ n=0 ∞ √((2n)!)/(n!)

I found that ρ_n = √(2n+1)/(n+1).

Then, I found ρ = lim when n→∞ |(1/n) (√(2n+1))/((1/n) (n+1))| = 0

Based on this result I concluded the series converges; however, the book answer says it diverges. What am I doing wrong?

 

[Mark44]

Homework Statement:: Use the ratio test to find whether the following series converge or diverge:
Homework Equations:: ∑ n=0 ∞ √((2n)!)/(n!)

I found that ρ_n = √(2n+1)/(n+1).

Then, I found ρ = lim when n→∞ |(1/n) (√(2n+1))/((1/n) (n+1))| = 0

Based on this result I concluded the series converges; however, the book answer says it diverges. What am I doing wrong?

For the ratio test you should have $$\lim_{n \to \infty} \frac{\sqrt{(2n + 2)!}}{(n + 1)!} \frac{n!}{\sqrt{(2n)!}}$$

I wrote the above as $a_{n+1} \cdot \frac 1 {a_n}$
The limit I get is not zero.

 

[Fernando Rios]

For the ratio test you should have $$\lim_{n \to \infty} \frac{\sqrt{(2n + 2)!}}{(n + 1)!} \frac{n!}{\sqrt{(2n)!}}$$

I wrote the above as $a_{n+1} \cdot \frac 1 {a_n}$
The limit I get is not zero.

Shouldn't it be (2n +1) instead of (2n+2)?

 

[Fernando Rios]

First, no.
No. 2(n+1)= 2n+ 2 so (2(n+1))!= (2n+ 2)!= (2n+2)(2n+ 1)(2n)(2n-1)... Dividing by (2n)! leaves (2n+2)(2n+1).

Thank you for your response. I see my mistake now.