MHB Solving Series Questions: Limit of $\frac{(2n-1)!}{(2n+1)!}$ & More

[annie122]

I have some series related questions:

1) Find the limit of $\frac{(2n-1)!}{(2n+1)!}$

I know each term is less than one, but i don't know how to use this to get the limit

2) Find the limit of the sequence $\left(\sqrt{2},\,\sqrt{2\sqrt{2}},\,\sqrt{2\sqrt{2 \sqrt{2}}},\cdots\right)$

3) $a_n = \sqrt{2 + a_n-1}$

a) show $(a_n)$ is increasing and bounded above by 3.
b) Find the limit of $a_n$.

I have no clue for the last two.

 

[alyafey22]

Note that

$$(2n+1)!= (2n+1) 2n (2n-1)!$$

The second question can be restated to find

$$a_{n+1}=\sqrt{2a_n }$$ where $a_0 =1$

 

[Evgeny.Makarov]

I have some series related questions:

Also please note that "series" is the sum of the terms of a sequence.

 

[alexmahone]

Are you sure the 3rd question is stated correctly?

$$a_n=\sqrt{2+a_n-1}$$?

 

[annie122]

thanks, i now got the answer for the first one.
but i still don't know how to approach the problem even if it's restated that way.

and yes, I'm sure i wrote it as it is in the textbook.

 

[alyafey22]

Hint:

Try induction.