Infinite Sequence Involving A Factorial

Bashyboy
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Homework Statement


a_n = \frac{(2n -1)!}{(2n)^n}


Homework Equations





The Attempt at a Solution


I am not exactly sure how to solve this problem.
 
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Bashyboy said:

Homework Statement


a_n = \frac{(2n -1)!}{(2n)^n}


Homework Equations





The Attempt at a Solution


I am not exactly sure how to solve this problem.


What are you supposed to do with an? Find its limit? Sum it?

RGV
 
Oh, I am sorry that I did not specify. I need to take the limit as n goes to infinity of this sequence.
 
Does it seem, to anyone, that I have left any more information out?
 
Hint: Try looking at a bound...what's it greater than, or what's it less than. Then take the limit of that, you should have your answer.
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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