# Convergent Subsequences....

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• Dec 19th 2009, 08:58 AM
zhupolongjoe
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 (Doh)

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.
• Dec 19th 2009, 09:16 AM
Plato
We can agree that $x_n = \frac{{\left( {1 + \frac{1}{n}} \right)^2 + \left( { - 1} \right)^n }}{{1 + \frac{1}{n}}}$.

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

$z_n = x_{2n - 1} = \frac{{\left( {1 + \frac{1}{{2n - 1}}} \right)^2 - 1}}{{1 + \frac{1}{{2n}}}} \to 0$
• Dec 19th 2009, 09:23 AM
zhupolongjoe
Oh I see...you just took the even terms and the odd terms. Makes sense, thanks.