# Thread: Convergent Subsequences....

1. ## Convergent Subsequences....

I am reviewing my analysis midterm to study for the final, and I can't seem to remember how to get the correct solution to one of the problems.

It says: consider the sequence {xn} with
{xn}=((n+1)^2+(-1)^n*n^2)/(n*(n+1))

Determine if it converges or diverges.

I got that it diverges, which is correct, but I didn't get credit because my method was invalid

I just did a simple divide through by the highest term and you see that it boils down to 1+(-1)^n, which diverges. Of course you can't actually do this because in doing so I took terms like 1/n^2 and said they converge to 0, and so, this is what you are left with.

The professor said the correct solution involves finding two subsequences which converge to different values. I am not sure how to do this. Any help please? Thanks.

2. We can agree that $\displaystyle x_n = \frac{{\left( {1 + \frac{1}{n}} \right)^2 + \left( { - 1} \right)^n }}{{1 + \frac{1}{n}}}$.

$\displaystyle y_n = x_{2n} = \frac{{\left( {1 + \frac{1}{{2n}}} \right)^2 + 1}} {{1 + \frac{1}{{2n}}}} \to 2$.

$\displaystyle z_n = x_{2n - 1} = \frac{{\left( {1 + \frac{1}{{2n - 1}}} \right)^2 - 1}}{{1 + \frac{1}{{2n}}}} \to 0$

3. Oh I see...you just took the even terms and the odd terms. Makes sense, thanks.