# Infinite Series (The Ratio Test)

• Fernando Rios
In summary, using the ratio test to determine if the series ∑ n=0 ∞ √((2n)!)/(n!) converges or diverges, I found that the limit as n approaches infinity is not equal to 0, indicating that the series actually diverges.
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 ρn = √(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?

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

I found that ρn = √(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.

Mark44 said:
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)?

HallsofIvy said:
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.

## 1. What is the Ratio Test for infinite series?

The Ratio Test is a method used to determine whether an infinite series converges or diverges. It involves finding the limit of the ratio of consecutive terms in the series. If the limit is less than 1, the series converges. If the limit is greater than 1 or infinite, the series diverges.

## 2. How do you use the Ratio Test to determine convergence?

To use the Ratio Test, you first find the limit of the ratio of consecutive terms in the series. If the limit is less than 1, the series converges. If the limit is greater than 1 or infinite, the series diverges. If the limit is equal to 1, the test is inconclusive and another method must be used to determine convergence.

## 3. What is the difference between absolute and conditional convergence?

Absolute convergence refers to a series that converges regardless of the order of the terms. Conditional convergence refers to a series that only converges when the terms are arranged in a specific order.

## 4. Can the Ratio Test determine the exact value of a convergent series?

No, the Ratio Test can only determine whether a series converges or diverges. To find the exact value of a convergent series, other methods such as the geometric series formula or Taylor series expansion must be used.

## 5. Are there any limitations to using the Ratio Test?

Yes, the Ratio Test can only be used for series with positive terms. It also cannot determine the convergence or divergence of alternating series or series with terms involving factorials.

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