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Let ${x_n}$ be the sequence $+\sqrt{1}, -\sqrt{1},+\sqrt{2}, -\sqrt{2},+\sqrt{3}, -\sqrt{3}$ ...


$$y_n = \frac{{x_1}+{x_2}+...+{x_n}}{n}$$ for all $n \in\Bbb N$, then the sequence $\{y_n\}$ is:

a) Monotonic or b) Not Bounded or c) Bounded but not Convergent or d) Convergent.

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What have you tried? It's hard to give hints here. Try to work by elimination. – Raskolnikov Aug 14 '12 at 6:04
Calculate $y_1$, $y_2$, $y_3$, $y_4$, $y_5$, $y_6$. (No calculaor, just observe the obvious cancellations.) I think the answers will become clear. – André Nicolas Aug 14 '12 at 6:08
This question was in my exam. Now,I can try to solve it ...thanks – ram Aug 14 '12 at 6:09
Well clearly a) is not the answer, since if the sequence is monotonic then it is either not bounded or convergent. – Alex Becker Aug 14 '12 at 7:04

If you don’t immediately see what’s going on, always try calculating $y_n$ for small $n$:

$$\begin{align*} y_1&=\sqrt 1=1\\ y_2&=0\\ y_3&=\frac{\sqrt2}3\\ y_4&=0\\ y_5&=\frac{\sqrt3}5\\ y_6&=0\\ y_7&=\frac{\sqrt4}7 \end{align*}$$

Look for patterns, and then try to see why they’re there. Here you shouldn’t have too much trouble spotting the patterns and writing down a general formula for $y_n$ when $n$ is even and another for $y_n$ when $n$ is odd. Even with the seven values that I’ve calculated here you should be able to explain why (a) is not the answer, and once you figure out the general formulas, the right answer is easy to pick out.

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I think, the ansewr must be (d), because $y_n:=\begin{cases} 0 & if\,n\,is\,even.\\ \frac{\sqrt {(n+1)/2}}{ n} & if\, n \,is\, odd.\\ \end{cases}$ $\implies\lim_{\,\,n\to\infty} y_n = 0.$ $\implies y_n$ converges to $0$. – ram Aug 14 '12 at 9:17
@ram: Yes, that does it just it just fine. (Sorry that I didn’t notice that you had the indexing wrong the first time.) – Brian M. Scott Aug 14 '12 at 9:19
ok...Sir. Thanks a lot. – ram Aug 14 '12 at 9:21

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