Checking the convergence of this numerical series using the ratio test

[DottZakapa]

Homework Statement

using ratio test verify if converges

Relevant Equations

convergence tests

$\sum_{n=0}^\infty \frac {(2n)!}{(n!)^2}$

$\lim_{n \rightarrow +\infty} {\frac {a_{n+1}} {a_n}}$

that becomes

$\lim_{n \rightarrow +\infty} {\frac { \frac {(2(n+1))!}{((n+1)!)^2}} { \frac {(2n)!}{(n!)^2}}}$

$\lim_{n \rightarrow +\infty} \frac {(2(n+1))!(n!)^2}{((n+1)!)^2(2n)!}$

$\lim_{n \rightarrow +\infty} \frac {(2n+2))!(n!)(n!)}{(n+1)!(n+1)!(2n)!}$

then i don't know what else i can do

 

[Gaussian97]

First of all, what does the ratio test say?

 

[DottZakapa]

First of all, what does the ratio test say?

$\lim_{n \rightarrow +\infty} {\frac {a_{n+1}} {a_n}}$

that becomes

$\lim_{n \rightarrow +\infty} {\frac { \frac {(2(n+1))!}{((n+1)!)^2}} { \frac {(2n)!}{(n!)^2}}}$

$\lim_{n \rightarrow +\infty} \frac {(2(n+1))!(n!)^2}{((n+1)!)^2(2n)!}$

$\lim_{n \rightarrow +\infty} \frac {(2n+2))!(n!)(n!)}{(n+1)n!(n+1)n!(2n)!}$

 

[Gaussian97]

Actually the ratio test uses the absolute values, in this case, all terms are positive so doesn't matter but is important to know exactly do the theorems say.

Ok, now using the properties of factorials, do you see any way to simplify this expression?

 

[DottZakapa]

Ok, now using the properties of factorials, do you see any way to simplify this expression?

that is the point, I've already applied all the factorial properties that are in my knowledge. if there are others could you please tell me them? I would really appreciate it :) .
i have always problems with factorials.
thanks

 

[Gaussian97]

Well, actually you only need the most fundamental property:
$$n! = n \cdot (n-1)!$$

 

[DottZakapa]

Well, actually you only need the most fundamental property:
$$n! = n \cdot (n-1)!$$

like this?
$\lim_{n \rightarrow +\infty} \frac {(2n+2)!}{(n+1)(n+1)2n(n-1)!}$

 

[Gaussian97]

Well, a little bit better. Be careful because $(2n)!\neq 2n\cdot(n-1)!$, and you can still use this property to further simplify you answer (you can actually get rid of all the factorial terms using this property)

 

[DottZakapa]

Well, a little bit better. Be careful because $(2n)!\neq 2n\cdot(n-1)!$, and you can still use this property to further simplify you answer (you can actually get rid of all the factorial terms using this property)

$2n(2n-2)! ?$

 

[Gaussian97]

$2n(2n-2)! ?$

No, use better the property in the form $$x! = x\cdot(x-1)!,$$ if you substitute $x=2n$ what do you obtain?

 

[DottZakapa]

No, use better the property in the form $$x! = x\cdot(x-1)!,$$ if you substitute $x=2n$ what do you obtain?

Very good! you've been super good, did not consider it.
thanks a lot.
Now it simplified as it should