# Thread: limit of a sequence

1. ## limit of a sequence

hi,
find the lim of |1/(((n+1)^2)*2^n)|^(1/n).

Sorry if that's a bit messy.
Thanks

2. Originally Posted by tbyou87
hi,
find the lim of |1/(( (n+1)^2 )*2^n)|^(1/n).

Sorry if that's a bit messy.
Thanks
$\displaystyle \lim \ \left( \frac{1}{2^n (n+1)^2} \right)^{1/n} = \lim \ \frac{1}{2(n+1)^{1/n}(n+1)^{1/n}} =$$\displaystyle \frac{1}{2n^{1/n} \left( 1 + \frac{1}{n} \right)^{1/n} \cdot n^{1/n} \left( 1 + \frac{1}{n} \right)^{1/n} } = \frac{1}{2e^2}$.

3. Hi,
When I plug in large values for n such as n=300 i get .48143 while the limit 1/2(e^2) is around .0676676.

I am doing something wrong?

4. No I am doing something wrong. I did the limit of e is $\displaystyle (1+1/n)^n$ NOT $\displaystyle (1+1/n)^{1/n}$. (I had something to drink).

But $\displaystyle (n+1)^{1/n} \to 1$ because:
$\displaystyle n^{1/n} \leq (n+1)^{1/n} \leq (n+n)^{1/n} = 2^{1/n} \cdot n^{1/n}$.
Use the squeze theorem.

5. The limit is thus 1/2 right?

6. Originally Posted by tbyou87
The limit is thus 1/2 right?
Why you asking me? It is clear that it is 1/2.