# Thread: sequences with factor'ials?

1. ## sequences with factor'ials?

confirm is this one divergent?

inf x!
sum ------------------------
x=1 1 * 3 * 5 ... ( 2 x -1)

I said that we should start by replacing all n with n+1

Then we multiply an+1 by the reciprocal of an

That leads me to a messy fraction.

I get all messed up with a fraction that looks like:

(x+1)(2x-1)
-----------
2x+1

I come up with the series is divergent but due to abs value of ratio an+1 to an.

How do you solve this so the fraction obtained after the reciprocal is one that is easily taken a limit of to determine divergence or convergence?

2. Originally Posted by orendacl
confirm is this one divergent?

inf x!
sum ------------------------
x=1 1 * 3 * 5 ... ( 2 x -1)

I said that we should start by replacing all n with n+1

Then we multiply an+1 by the reciprocal of an

That leads me to a messy fraction.

I get all messed up with a fraction that looks like:

(x+1)(2x-1)
-----------
2x+1

I come up with the series is divergent but due to abs value of ratio an+1 to an.

How do you solve this so the fraction obtained after the reciprocal is one that is easily taken a limit of to determine divergence or convergence?
Ratio test.

$\lim_{n \to \infty}\bigg{|} \frac{a_{n+1}}{a_n} \bigg{|}$

$= \lim_{n \to \infty}\bigg{|} \frac{(x+1)! \cdot (1 \cdot 3 \cdot 5 \cdot \dots \cdot (2x-1)}{x! \cdot (1 \cdot 3 \cdot 5 \cdot \dots \cdot (2x+1)} \bigg{|}$

$= \lim_{n \to \infty}\bigg{|} \frac{x+1}{2x(2x+1)} \bigg{|}$

$= \lim_{n \to \infty}\bigg{|} \frac{x+1}{4x^2 + 2x} \bigg{|} < 1$. Hence converges.

In fact, I think it converges to...

$1 + \frac{\pi}{2}$...